
tesQB^ 



Glass. 



AN 



ELEMENTARY TREATISE s<9 



ASTRONOMY: 

IN FOUR PARTS. 



CONTAINING 



A SYSTEMATIC AND COMPREHENSIVE EXPOSITION OF THE THEORY, 
AND THE MORE IMPORTANT PRACTICAL PROBLEMS : 



SOLAR, LUNAR, AND OTHER ASTRONOMICAL TABLES, 



DESIGNED FOR USE AS A 

TEXT-BOOK IN COLLEGES AND THE HIGHER ACADEMIES. 



BY 

WILLIAM A. NORTON, A. M., 

FELLOW OF THE AMERICAN PHILOSOPHICAL SOCIETY, AND OF THE AMERICAN 

ACADEMY OF ARTS AND SCIENCES, AND CORRESPONDING 

MEMBER OF THE NATIONAL INSTITUTE. 



THIRD EDITION. 

CORRECTED, IMPROVED, AND ENLARGED. 



NEW YORK: 
JOHN WILEY, 107 BROADWAY. 



1852. 






L*-- iff 



Entered according to Act of Congress, in the year 1845. 

By WILLIAM A NORTON, 

In the Clerk's Office of the District Court for the Southern District of New York. 



Stereotyped by 
RICHARD C. VALENTINE. 

17 Dutch-street, New YorK . 



PREFACE 

TO THE FIRST EDITION. 



The object for which the present treatise on Astronomy has 
been written, is to provide a suitable text-book for the use of the 
students of Colleges and the higher Academies, and at the same 
time to furnish the practical astronomer with rules, or formulae, 
and accurate tables for performing the more important astronomi- 
cal calculations. 

It is divided into four Parts. The first three Parts contain the 
theory : the First Part treating of the determination of the places 
and motions of the heavenly bodies ; the Second, of the phenom- 
ena resulting from the motions of these bodies, and of their ap- 
pearances, dimensions, and physical constitution ; and the Third, 
of the theory of Universal Gravitation. The Fourth Part consists 
of Practical Problems, which are solved with the aid of the Tables 
appended to the work. An Appendix is added, containing a large 
collection of useful trigonometrical formulae, and such investiga- 
tions of astronomical formulae as, from their length, could not, 
consistently with the plan of the work, be admitted into the text, 
and which it was still deemed advisable to retain, for the benefit 
of the few who might wish to pursue them. 

The chief peculiarities of this treatise, as compared with the 
kindred works now in use in our Colleges, are, — 1. The adoption 
of the Copernican System as an hypothesis at the outset, leaving 
it to be established by the agreement between the conclusions to 
which it leads and the results of observation. 2. A connected ex- 
position of the principles and methods of astronomical observation, 
embracing the doctrine of the sphere, the construction and use of 



IV PREFACE TO THE FIRST EDITION. 

the principal astronomical instruments, and the theory of the cor- 
rections for refraction, parallax, aberration, precession, and nuta- 
tion. 3. The exhibition of the methods of determining the motions 
and places of the different classes of the heavenly bodies, in one 
connection. 4. The explanation of the principles of the construc- 
tion of astronomical tables. 5. The addition of a chapter on the 
measurement of time, embracing the explanation of the different 
kinds of time, the processes by which one is converted into an- 
other, the methods of determining the time from astronomical obser- 
vations with the transit instrument and sextant, and the calendar. 
6. The contemplation of the phenomena of the aspect and appa- 
rent motion of the heavenly bodies as consequences of their motions 
in space, and the deduction of the various circumstances of these 
phenomena from the theory of the orbitual motions previously es- 
tablished. 7. A comprehensive view of the theory of Universal 
Gravitation, followed out into its various consequences. 8. An 
exposition of the operations of the disturbing forces in producing 
the principal perturbations of the motions of the Solar System. 
9. The solution of Practical Problems by means of logarithmic 
formulae instead of rules. 10, The addition of lunar, solar, and 
other astronomical tables, of peculiar accuracy and improved ar- 
rangement. 

It may further be mentioned, that many of the investigations 
have been materially simplified, and that the aim has been to in- 
troduce into all of them as much simplicity and uniformity of 
method as possible. Particular attention has also been paid to 
the diagrams, it being of great importance that they should convey 
correct notions to the mind of the student. 

The Problems in the Fourth Part are principally for making 
calculations relative to the Sun, Moon, and Fixed Stars. The 
Tables of the Sun and Moon, used in finding the places of these 
bodies, have, for the most part, been abridged and computed from 
the tables of Delambre, as corrected by Bessel, and those of 
Burckhardt ; and the Tables of Epochs have all been reduced to 
the meridian of Greenwich. These Tables will give the places 



PREFACE TO THE FIRST EDITION. V 

and motions of the Sun and Moon within a fraction of a second of 
the tables from which they were derived. But as this degree of 
accuracy will not generally be required, rules are also given in 
the Fourth Part for obtaining approximate results. The entire set 
of Tables has been stereotyped, and great pains has been taken, 
by repeated revisions and verifications, to render them accurate. 

The principal astronomical works which have been consulted, 
are those of Vince, Gregory, Woodhouse, Delambre, Biot, La 
place, Herschel, and Gummere ; also Francozur's Uranography, 
Francozur's Practical Astronomy, Encyclopedia Metropolitana, 
Article " Astronomy," and Baily's Tables and Formula. Free 
use has been made of the methods of investigation and demonstra- 
tion pursued in these treatises, such modifications being intro- 
duced, in those which have been adopted, as the plan of the work 
required. 

New York, January, 1839. 



PREFACE 

TO THE SECOND EDITION. 



In preparing a new edition of the present treatise, material al- 
terations, and, it is hoped, improvements have been made in it. 
The more abstruse parts are now printed in smaller type, and 
their connection with the other portions of the book is made such 
that they can be pursued or omitted at pleasure : by- which the 
opportunity is afforded of making a selection between two courses 
of study, differing materially in extent, and in the amount of labor 
and mathematical attainment required for their acquisition. Wood- 
cuts have also been substituted for the original plates, as more 
convenient to the student ; and for the sake of more ample illus- 
tration, nearly fifty new diagrams have been added. Many of 
these are illustrative of the telescopic appearances of the planets 
and other heavenly bodies. Considerable additions have been 
made to several of the Chapters ; especially to the Chapter on 
Instruments, and those in which the appearances and physical 
constitution of the heavenly bodies are treated of. These are, for 
the most part, printed in a small-sized type, as well as the parts 
above specified. The Chapters on Comets have been rewritten. 
The Author has also endeavored, in many instances which need 
not be enumerated, to profit by such criticisms and suggestions of 
improvement as have been made by others, as well as by his own 
experience in the use of the work as a text-book. 

The Tables remain unaltered ; with the exception of Tables I., 
II., III., and IV., which have been rendered more accurate. Fre- 
quent comparisons, since the publication of the first edition, of the 
Lunar and Solar Tables with the places of the Moon and Sun, as 



PREFACE TO THE SECOND EDITION. Vll 

given in the Nautical Almanac and the Connaissance des Terns, 
have furnished additional confirmation of their accuracy. 

Notwithstanding the considerable augmentation which the work 
has received, the retail price of it is very much reduced. 

The references in the text to the investigations of astronomical 
formulae in the Appendix, were omitted, -in preparing this edition, 
under the expectation that the new matter to be inserted would 
render the omission of these investigations necessary. They are, 
however, retained ; and the articles are designated in which men- 
tion is made of such formulae. 

In addition to the Astronomical works mentioned in the preface 
to the first edition, the Author has particularly consulted, in the 
preparation of this edition, besides periodicals, Littroiv's Wonders 
of the Heavens, Kendall's Uranography, NichoVs Phenomena of 
the Solar System, NichoVs Architecture of the Heavens, and Ma- 
son's Introduction to Practical Astronomy. His acknowledgments 
are due to Professor Kendall for the copy which he was permitted 
to take of the delineation of the great comet of 1843, given in his 
Uranography. 

Where passages have been borrowed entire from any author, 
credit has been given in the usual way, viz., by references to 
specifications of title, &c., inserted at the bottom of the page. 
To these it should be added that the greater portion of the Chap- 
ter on the Calendar, after the first paragraph, is taken from Wood- 
house's Astronomy, and most of Art. 463, from Gregory's Astron- 
omy. Particular assistance has also been derived, in Part IV., 
from Gummere's Astronomy. It would be idle in every new 
scientific treatise, to attempt to designate all the instances in which 
the same forms of expression and the same methods of investiga- 
tion may have been adopted, that occur in other kindred treatises. 
Delaware College, 

Newark, Del., June, 1845. 



PREFACE 

TO THE THIRD EDITION. 



Since the publication of the previous edition, numerous im- 
portant and highly interesting astronomical discoveries have 
been made. These have been introduced into the present 
edition, by appending a collection of Notes to the text. The 
references to these notes, inserted in the text, will bring the 
different topics of which they treat to the notice of the stu- 
dent, in the proper connection, while they will collectively 
form a brief exposition of the progress recently made in astro- 
nomical science. It has been the intention to make this edi- 
tion a faithful picture of the present state of the science ; in 
so far as this end could be attained within the limits which 
should be observed in the preparation of a college text-book, 

Providence, Ap»l, 1852. 



TABLE OF CONTENTS. 



INTRODUCTION. 



Paok 

General Notions — General Phenomena of the Heavens 1 



PART I 

ON THE DETERMINATION OF THE PLACES AND MOTIONS OF 
THE HEAVENLY BODIES. 

CHAPTER I. 
On the Celestial and Terrestrial Spheres - - - - - 11 

CHAPTER II. 

On the Construction and Use of the Principal Astronomical Instruments 23 
Transit Instrument -------- 26 

Astronomical Clock - - - - - - - - 31 

Astronomical Circle -- = .----32 

Altitude and Azimuth Instrument ------ 35 

Equatorial ------- v - _ 37 

Sextant -----.,-.-39 

Errors of Instrumental Admeasurement ----- 42 

Telescope - - - %b. 

CHAPTER III. 

On the Corrections of the Co-ordinates of the Observed Place of a 

Heavenly Body 43 

Refraction --- 44 

Parallax --- 49 

Aberration -------.-.55 

Precession and Nutation ------- 60 

Remarks on the Corrections. — Verification of the Hypothesis 

that the Diurnal Motion of the Stars is Uniform and Circular 65 

2 



TABLE OF CONTENTS. 



CHAPTER IV. 

Pag« 

Of the Earth — its Figure and Dimensions — Latitude and Longitude 
of a Place ------ ^ - - - - 06 

Determination of the Latitude and Longitude of a Place 69 



CHAPTER V. 
Of the Places of the Fixed Stars 72 

CHAPTER VI. 

Of the apparent motion of the Sun in the Heavens - 77 

CHAPTER VII. 

Of the Motions of the Sun, Moon, and Planets, in their orbits - 82 

Kepler's Laws - - ib. 

Definitions of Terms - - - - - - - -85 

Elements of the Orbit of a Planet 87 

Methods of Determining the Elements of the Sun's Apparent 

Orbit, or of the Earth's Real Orbit 88 

Methods of Determining the Elements of the Moon's Orbit - 92 

Methods of Determining the Elements of a Planet's Orbit - 94 

Mean Elements and their Variations - - - - - 102 

CHAPTER VIII. 

On the Determination of the Place of a Planet, or of the Sun, or 
Moon, for a Given Time, by the Elliptical Theory ; and of the 

Verification of Kepler's Laws - - - - - - 105 

Place of a Planet, or of the Sun or Moon in its Orbit - - ib. 

Heliocentric Place of a Planet ---_-_ J06 

Geocentric Place of a Planet - - 107 

Places of the Sun and Moon - - - -. - - 108 

Verification of Kepler's Laws --.._._ iff, 

CHAPTER IX. 

On the Inequalities of the Motions of the Planets and of the Moon ; 
and of the Construction of Tables for finding the Places of these 

Bodies ------109 

Construction of Tables - -_ _ . _ . 114 



TABLE OF CONTENTS. X\ 



CHAPTER X. 



Pag» 

Of the Motions of the Comets - - - . • ^ 117 



CHAPTER XL 
Of the Motions of the Satellites 124 



CHAPTER XIL 

On the Measurement of Time ------- 127 

Different Kinds of Time ------- $. 

Conversion of one Species of Time into another - - - 128 
Determination of the Time and Regulation of Clocks by Astro- 
nomical Observations - - - - - - - 130 

Of the Calendar - - - - - - - - - 133 



PART II. 

ON THE PHENOMENA RESULTING FROM THE MOTIONS OF 
THE HEAVENLY BODIES, AND ON THEIR APPEARANCES, 
DIMENSIONS, AND PHYSICAL CONSTITUTION. 

CHAPTER XIII. 

Oi the Sun and the Phenomena attending its Apparent Motions - 137 

Inequality of Days --------- ib. 

Twilight - ■ 141 

The Seasons --------- 144 

Appearance, Dimensions, and Physical Constitution of the Sun 147 

CHAPTER XIV. 

Of the Moon and its Phenomena - - - - - - 153 

Phases of the Moon „.- ib 

Moon's Rising, Setting, and Passage over the Meridian - 155 

Rotation and Librations of the Moon - - - - - 158 

Dimensions and Physical Constitution of the Moon - - 159 



XU TABLE OF CONTENTS . 



CHAPTER XV. 



Eclipses of the Sun and Moon — Occultations of the Fixed Stars - 162 

Eclipses of the Moon -------'- ib. 

Eclipses of the Sun - - - --- - - - 171 

Occultations _ - . - 183 



CHAPTER XVI. 

Of the Planets and the Phenomena occasioned by their Motions in 

Space • 184 

Apparent Motions of the Planets with respect to the Sun - ib. 

Stations and Retrogxadations of the Planets - - - - 187 

Phases of the Inferior Planets ------ 190 

Transits of the Inferior Planets -*.--- 191 
Appearances, Dimensions, Rotation, and Physical Constitution 

of the Planets - * 192 



CHAPTER XVIL 

Of Comets - 201 

Their General Appearance — Varieties of Appearance ib. 

Form, Structure, and Dimensions of Comets - 205 

Physical Constitution of Comets - 207 

CHAPTER XVIII. 

Of the Fixed Stars -------.,- 211 

Their Number and Distribution over the Heavens - - ib. 

Annual Parallax and Distance of the Stars - 213 

Nature and Magnitude of the Stars 216 

Variable Stars - - - 5>J7 

Double Stars - 219 

Proper Motions of the Stars 222 

Clusters of Stars — Nebulae 223 

Distance and Magnitude of Nebulas ... 227 

Structure of the Material Universe — Nebular Hypothesis - 229 



TABLE OF CONTENTS. XUl 

PART III. 

OF THE THEORY OF UNIVERSAL GRAVITATION. 

CHAPTER XIX. 

Page 
Of the Principle of Universal Gravitation - - - - - 231 

CHAPTER XX. 

Theory of the Elliptic Motion of the Planets - 234 

CHAPTER XXL 

Theory of the Perturbations of the Elliptic Motion of the Planets 

and of the Moon 239 

CHAPTER XXIL 

Of the Relative Masses and Densities of the Sun, Moon, and Planets ; 

and of the Relative Intensity of the Gravity at their surface - 249 

CHAPTER XXIIL 

Of the Figure and Rotation of the Earth ; and of the Precession of 

the Equinoxes and Nutation .__-_- 251 

CHAPTER XXIV. 
Of the Tides 254 



PART IV. 

ASTRONOMICAL PROBLEMS. 

Explanations of the Tables -----,- 261 

Prob. I. To work, by logistical logarithms, a proportion the terms 
of which are degrees and minutes, or minutes and seconds, of 
an arc ; or hours and minutes, or minutes and seconds, of time 266 



XIV TABLE OF CONTENTS. 



Pags 



Prob. II. To take from a table the quantity corresponding to a 
given value of the argument, or to given values of the arguments 
of the table 267 

Prob. III. To convert Degrees. Minutes, and Seconds of the Equa- 
tor into Hours, Minutes, &c, of Time - 273 

Prob. IV. To convert Time into Degrees, Minutes, and Seconds ib. 

Prob. V. The Longitudes of two Places, and the Time at one of 

them being given, to find the corresponding time at the other 274 

Prob. VI. The Apparent time being given, to find the correspond- 
ing Mean Time ; or, the Mean Time being given, to find the 
Apparent ---------- 275 

Prob. VII. To correct the Observed Altitude of a Heavenly Body 

for Refraction 278 

Prob. Till. The Apparent Altitude of a Heavenly Body being 

given, to find its True Altitude - - - - - - 279 

Prob. IX. To find the Sun's Longitude, Hourly Motion, and Semi- 
diameter, for a given Time, from the Tables - 281 

Prob. X. To find the Apparent Obliquity of the Ecliptic, for a 

given Time, from the Tables ------ 283 

Prob. XL Given the Sun's Longitude and the Obliquity of the 

Ecliptic, to find his right Ascension and Declination - - 284 

Prob. XII. Given the Sun's Right Ascension and the Obliquity of 

the Ecliptic, to find his Longitude and Declination - - 285 

Prob. XIII. The Sun's Longitude and the Obliquity of the Ecliptic 

being given, to find the Angle of Position - 285 

Prob. XIV. To find from the Tables, the Moon's Longitude, Lati- 
tude, Equatorial Parallax, Semi-diameter, and Hourly Motions 
in Longitude and Latitude, for a given Time - 286 

Prob. XV. The Moon's Equatorial Parallax, and the Latitude of 

a Place, being given, to find the Reduced Parallax and Latitude 295 

Prob." XVI. To find the Longitude and Altitude of the Xonagesi- 

mal Degree of the Ecliptic, for a given Time and Place - ib, 

Prob. XVII. To find the Apparent Longitude and Latitude, as 
affected by Parallax, and the Augmented Semi-diameter of the 
Moon ; the Moon's True Longitude, Latitude, Horizontal Semi- 
diameter, and Equatorial Parallax, and the Longitude and Alti- 
tude of the Nonagesimal Degree of the Ecliptic, being given 298 

Prob. XVIII. To find the Mean Right Ascension and Declination, 
or Longitude and Latitude of a Star, for a given Time, from the 
Tables 302 



TABLE OF CONTENTS. XV 

Page 
Prob. XIX. To find the Aberrations of a Star in Right Ascension 

and Declination, for a given Day - 303 

Prob. XX. To find the Nutations of a Star in Right Ascension 

and Declination, for a given Day - 304 

Prob. XXI. To find the Apparent Right Ascension and Declina- 
tion of a Star, for a given day ___-.- 306 

Prob. XXII. To find the Aberrations of a Star in Longitude and 

Latitude, for a given Day - - - - - - - 307 

Prob. XXIII. To find the Apparent Longitude and Latitude of a 

Star, for a given Day ------- ib. 

Prob. XXIV. To compute the Longitude and Latitude of a 
Heavenly Body from its Right Ascension and Declination, the 
Obliquity of the Ecliptic being given ----- 308 

Prob. XXV. To compute the Right Ascension and Declination 
of a Heavenly Body from its Longitude and Latitude, the 
Obliquity of the Ecliptic being given - 309 

Prob. XXVI. The Longitude and Declination of a Body being 
given, and also the Obliquity of the Ecliptic, to find the Angle 
of Position 310 

Prob. XXVII. To find from the Tables the Time of New or Full 

Moon, for a given Year and Month - - - - - 311 

Prob. XXVIII. To determine the number of Eclipses of the Sun 
and Moon that may be expected to occur in any given Year, 
and the Times nearly at which they will take place - - 314 

Prob. XXIX. To calculate an Eclipse of the Moon - 317 

Prob. XXX. To calculate an Eclipse of the Sun, for a given Place 321 

Prob. XXXI. To find the Moon's Longitude, &c, from the Nau- 
tical Almanac 338 



APPENDIX. 

Trigonometrical Formula - 341 

I. Relative to a Single arc or angle a - - ib. 
n. Relative to Two Arcs a and b, of which a is supposed to be 

the greater --------- ib. 

III. Trigonometrical Series ------ 343 

IV. Differences of Trigonometrical Lines - ib. 

V. Resolution of Right-angled Spherical Triangles - - ib. 

VI. Resolution of Oblique-angled Spherical Triangles - - 345 



XVI TABLE OF CONTENTS. 

Page 
Investigation of Astronomical Formula - 348 

Formulae for the Parallax in Right Ascension and Declination, 

and in Longitude and Latitude - ib. 

Formulas for the Aberration in Longitude and Latitude, and in 

Right Ascension and Declination - 355 

Formulae for the Nutation in Right Ascension and Declination 359 

Formulae for computing the effects of the Oblateness of the 

Earth's Surface, upon the Apparent Zenith Distance and 

Azimuth of a Star 363 

Solution of Kepler's Problem, by which a Body's Place is found 

in an Elliptical Orbit 364 



JSTOTES. 
LtoXXIL 369-382 



AN 



ELEMENTARY TREATISE 

ON 

ASTRONOMY. 



INTRODUCTION. 

GENERAL NOTIONS GENERAL PHENOMENA OF THE HEAVENS. 

1 . The space, indefinite in extent, which is exterior to the earth, 
is called the Heaven or Heavens, or the Firmament. The sun, 
moon, and stars, the luminous bodies which are posited in this 
space, are called the Heavenly Bodies. The entire assemblage 
of these bodies is frequently called the Heavens. 

2. The most casual observation shows us that the heavenly 
bodies are subject to a variety of motions, as well as to various 
changes of appearance. The science which treats of the laws 
and causes of these motions and changes, is called Astronomy ;* 
or, more particularly, Astronomy is a mixed mathematical science, 
which treats of the motions, positions, distances, appearances, mag- 
nitudes, and physical constitution of the heavenly bodies. It has 
been divided into the two departments of Plane or Pure Astronomy, 
and Physical Astronomy. Plane Astronomy comprehends, 1st, the 
description of the motions, appearance, and structure of the 
heavenly bodies, and the description and explanation of their phe- 
nomena, which may be called Descriptive Astronomy ; 2d, the 
methods of observation and calculation employed in obtaining a 
knowledge of the facts embodied in Descriptive Astronomy, and 
the computation from these of the details of occasional phenome- 
na, as eclipses of the sun and moon, occultations of the stars, &c, 
which is denominated Practical Astronomy. Physical Astronomy 
investigates inductively the physical causes of the observed mo- 
tions and constitution of the great bodies of the material universe, 
and deduces, as a mechanical problem, from the one great cause, 
the principle of universal gravitation, all the minutiae of the celes- 
tial mechanism. 

* From AffTijp, a star, and vo/ioj, a law. 
1 



2 INTRODUCTION, 

3. The origin of the science of Astronomy is involved in obscurity; but It w 
supposed that its first truths were discovered in the early ages of the world by 
shepherds, who, at the same time, watched their flocks by night, and followed the 
motions and noted the varying aspects of the heavenly bodies. Each successive 
age, from that time to the present, has, with occasional interruptions, brought to it 
its contributions of observations and discoveries. The imposing character of the 
celestial phenomena, and their intimate relations to the every-day wants of life, as 
well as the superstitions of the ignorant, have, from time immemorial, conspired 
to attract to this science the interested attention of mankind, and promote its ad- 
vancement. From the very nature of things, some of its truths have only unfolded 
themselves, as century after century has passed away ; while others still await the 
lapse of future ages. Its history, in a theoretical point of view, presents two prom- 
inent epochs, viz : 1. The epoch of the discovery of the true system of the world, by 
Copernicus, towards the middle of the sixteenth century ; soon followed by the 
discovery of the exact laws of its motions in space, by Kepler, (early in the sev- 
enteenth century ;) which has so completely changed the whole face of the science, 
and has been succeeded by such a mass of observations of greatly increased accu- 
racy, and such an uninterrupted series of important discoveries, that it may 
almost be eaid to be the date of its origin, as the science is now taught. 2. The 
epoch of the discovery of universal gravitation, by Sir Isaac Newton, (1683;) a 
discovery that has brought Astronomy within the province of Mechanical Philoso- 
phy, and contributed greatly to its advancement and extension, by making known 
its physical theory, which has been developed by Laplace and others with great 
minuteness of detaih Contemplating the scien-ce from a practical point of view, 
we find that its most prominent epoch is that of the discovery of the telescope, a 
the beginning of the seventeenth century, since which time, by the adaptation of 
the telescope to instruments for admeasurement, and the improvement of these 
instruments, its means of research have been gradually perfected and extended, 
as art and science have advanced fcand in hand : until from a few shepherds, un- 
der the open sky on the plains of Chaldea, with naught but their natural powers- 
of vision, there has come to be a large body of professed Astronomers in charge of 
permanent observatories erected in almost every civilized country on the globe ; 
and furnished at the same time with telescopes that bring the heavenly bodies hun- 
dreds or even thousands of times nearer, and disclose a new world of celestial ob- 
jects, and with instruments that mark out, with the greatest precision, the ever 
varying places of all these bodies. 

4, To be able to form correct notions of the phenomena 
of the heavens, it is necessary to know the form of the earth. 
We learn from the following circumstances that the earth is 
a body of a globular form, insulated in space. 1st. When a 
vessel is receding from the land, an observer stationed upon 

Fig. 1. the coast, first loses sight 

^ of the hull, then of the 

^^^^^^ come interposed between the 



GENERAL NOTIONS. 3 

hull and the lower parts of the sails of a distant vessel, and the eye 
of the observer, if the sea were really what it appears to be, an 
indefinitely extended plane. 2d. At sea the visible horizon, or the 
line bounding the visible portion of. the earth's surface, is every- 
where a circle, of a greater or less extent, according to the altitude 
of the point of observation, and is on all sides equally depressed. 
To illustrate this proof, let BOA (Fig. 2) represent a portion 

Fig. 2. 




of the earth's surface supposed to be spherical, P the position of 
the eye of the observer, and DPC a horizontal line. If we 
conceive lines, such as PA and PB, to be drawn through the 
point of observation P, tangent to the earth in every direction, it is 
plain that these lines will all touch the earth at the same distance 
from the observer, and therefore that the line AGB, conceived to 
be traced through all the points of contact, A,B, &c, which would 
be the visible horizon, is a circle. It is also manifest that the an- 
gles of depression CPA, DPB, &c, of the horizon in different 
directions, would be equal, and that a greater portion of the earth's 
surface would be seen, and thus that the horizon would increase in 
extent, in proportion as the altitude of the point of observation, P, 
increased. 3d. Navigators, as it is well known, have sailed around 
the earth in different directions. These facts prove the surface of 
the sea to be convex, and the surface of the land conforms very 
nearly to that of the sea ; for the elevations of the highest moun- 
tains bear an exceedingly small proportion to the dimensions of the 
whole earth. 

5. If an indefinite number of lines, PA, PB, &c, be conceived 
to be drawn through the point of observation P, (Fig. 2,) touching 
the earth on all sides, a conical surface will be formed, having its 
vertex at P, and extending indefinitely into space. All heavenly 
bodies, which at any time are situated below this surface, have the 
earth interposed between them and the eye of the observer, and 
therefore cannot be seen. All bodies that are above this surface, 
which send sufficient light to the eye, are visible. That portion 



4 INTRODUCTION. 

of the heavens which is above this surface, presents the appear- 
ance of a solid vault or canopy, resting upon the earth at the visi- 
ble horizon, (see Fig. 2 ;) and to this vault the sun, moon, and stars 
seem to be attached. It is hardly necessary to remark that this is 
an optical illusion. It will be seen in the sequel that the heavenly 
bodies are distributed through space at various distances from the 
<earth, and that the distances of all of them are very great in com- 
parison with the dimensions of the earth. 

It will suffice, in the conception of phenomena, to suppose the 
-eye of the observer to be near the earth's surface, and that the 
imaginary conical surface above mentioned, which separates the 
visible from the invisible portion of the heavens, is a horizontal 
plane, confounded for a certain distance with the visible part of the 
earth. This is called the plane of the horizon, and sometimes the 
horizon simply. 

6. Up and down, at any place on the earth's surface, are from 
and towards the surface ; and thus at different places have every 
variety of absolute direction in space. 

This fact should not merely be acknowledged to be true, but should be dwelt 
upon until the mind has become familiarized to the conception of it, and divested, 
as far as possible, of the notion of an absolute up and down in space. 

7. The earth is surrounded with a transparent gaseous medium, 
called the earth's atmosphere, estimated to be some fifty miles in 
height ; through which all the heavenly bodies are seen. The at- 
mosphere is not perfectly transparent, but shines throughout with 
light received from the heavenly bodies, and reflected from its par- 
ticles ; and thus forms a luminous canopy over our heads by day 
and by night. This is called the sky. It appears blue because 
this is the color of the atmosphere ; that is, because the particles 
of the atmosphere reflect the blue rays more abundantly than any 
other color. By day the portion of the atmosphere which lies 
above the horizon is highly illuminated by the sun, and shines with 
so strong a light as to efface the stars. 

8. The most conspicuous of the celestial phenomena, is a con 
tinual motion common to all the heavenly bodies, by which they 
are carried around the earth in regular succession. The daily 
circulation of the sun and moon about the earth is a fact recog- 
nised by all persons. If the heavens be attentively watched on 
any clear evening, it will soon be seen that the stars have a motion 
precisely similar to that of the sun and moon. To describe the 
phenomenon in detail, as witnessed at night : — if, on a clear night, 
we observe the heavens, we shall find that the stars, while they 
retain the same situations with respect to each other, undergo a 
continual change of position with respect to the earth. Some will 
be seen to ascend from a quarter called the East, being replaced 
by others that come into view, or rise ; others, to descend towards 
the opposite quarter, the West, and to go out of view, or set : and 
if our observations be continued throughout the night, with the 



GENERAL PHENOMENA OF THE HEAVENS. 5 

east on our left, and the west on our right, the stars which rise in 
the east will be seen to move in parallel circles, entirely across the 
visible heavens, and finally to set in the west. Each star will 
ascend in the heavens during the first half of its course, and de- 
scend during the remaining half. The greatest heights of the 
several stars will be different, but they will all be attained towards 
that part of the heavens which lies directly in front, called the 
South. If we now turn our backs to the south, and direct our 
attention to the opposite quarter, the North, new phenomena will 
present themselves. Some stars will appear, as before, ascending, 
reaching their greatest heights, and descending ; but other stars 
will be seen, farther to tne north, that never set, and which appear 
to revolve in circles, from east to west, about a certain star that 
seems to remain stationary. This seemingly stationary star is 
called the Pole Star; and those stars that revolve about it, and nevej 
set, are called Circumpolar Stars. It should be remarked, how 
ever, that the pole star, when accurately observed by means of 
instruments, is found not to be strictly stationary, but to describe 
a small circle about a point at a little distance from it as a fixed 
centre. This point is called the North Pole., It is, in reality, 
about the north pole, as thus defined, and not the pole star, that 
the apparent revolutions of the stars at the north are performed. 
At the corresponding hours of the following night the aspect of the 
heavens will be the same, from which it appears that the stars re- 
turn to the same position once in about 24 hours. It would seem, 
then, that the stars all appear to move from east to west, exactly 
as if attached to the concave surface of a hollow sphere, which 
rotates in this direction about an axis passing through the station 
of the observer and the north pole of the heavens, in a space of 
time nearly equal to 24 hours. For the sake of simplicity this 
conception is generally adopted. This motion, common to all the 
heavenly bodies, is called their Diurnal Motion.. It is ascertained, 
by certain accurate methods of observation and computation, here- 
after to be exhibited, that the diurnal motion of the stars is strictly 
uniform and circular. 

9. It is important to observe, that the conception of a single sphere 
to which the stars are supposed to be attached, will not represent 
their diurnal motion, as seen from every part of the earth's, surface, 
unless the sphere be supposed to be of such vast dimensions that 
the earth is comparatively but a mere point at its centre.* 

10. A circle cut out of the heavens conceived to be a rotating 
sphere, by a plane passing through the axis of rotation, has a north 

* The student should strive to familiarize his mind with this notion of the sphere 
of the heavens. The disposition, so natural to every one, to conceive the stars to 
be at no very great distance from the earth, in comparison with the dimensions of 
so large a body, will, until it is overcome, often give rise to very erroneous concep- 
tions of the different appearances of the same phenomenon, as viewed from different 
points of the earth's surface 



6 INTRODUCTION. 

and south direction ; and a circle cut out by a plane perpendicular 
to the axis, has an east and west direction. 

11. The greater number of the stars preserve constantly the 
same relative position with respect to each other ; and they are 
therefore called Fixed Stars. There are, however, a few stars, 
called Planets* which are perpetually changing their places in the 
heavens, The number of the planets is ten. Each has a distinc- 
tive name, as follows : Mercury, Venus, Mars, Jupiter, Saturn, 
Uranus, Ceres, Pallas, Juno, and Vesta. Mercury, Venus, Mars, 
Jupiter, and Saturn are visible to the naked eye, and have been 
known from the most ancient times. The other five, namely, Ura- 
nus, Ceres, Pallas, Juno r and Vesta, cannot be seen without the 
assistance of the telescope, and were discovered by modern ob- 
servers.! (See Note I, at the end of the Appendix.) 

12. The planets are distinguishable from each other either by a dif- 
ference of aspect, or by a difference of apparent motion with respect 
to the sun. Venus and Jupiter are the two most brilliant planets : 
they are quite similar in appearance, but their apparent motions 
with respect to the sun are very different. Venus never recedes 
beyond 40° or 50° from the sun, while Jupiter is seen at every va- 
riety of angular distance from him. Mars is known by the ruddy 
color of his light. Saturn has a pale, dull aspect. 

13. The apparent motion of the planets is generally directed 
towards the east ; but they are occasionally seen moving towards 
the west. As their easterly prevails over their westerly motion, 
they all, in process of time, accomplish a revolution around the 
earth. The periods of revolution are different for each planet. 

14. The Sun and Moon are also continually changing their 
places among the fixed stars. 

15. From repeated examinations of the situation of the moon 
among the stars, it is found that she has with respect to them a 
progressive circular motion from toest to east, and completes a re- 
volution around the earth in about 27 days. 

16. The motion of the sun is also constantly progressive, and 
directed from luest to east. This will appear on observing for a 
number of successive evenings the stars which first become visi- 
ble in that part of the heavens where the sunsets. It will be found 
that those stars which in the first instance were observed to set 
just after the sun, soon cease to be visible, and are replaced by 
others that were seen immediately to the east of them ; and that 
these, in their turn, give place to others situated still farther to the 
east. The sun, then, is continually approaching the stars that lie 

* Fvom jthainrmSi a wanderer. 

t The planet Uranus was discovered in 1781 by Dr. Herschel, who gave it the 
name of Georgium Sidus. By the European astronomers it was called Herschel. 
It is now generally known by the name given in the test. Ceres, Pallas, Juno, 
and Vesta have been discovered since 1600 ; the first by Piazzi, the second and 
fourth by Olbers, and the third by Harding. 



GENERAL PHENOMENA OF THE HEAVENS. 




on the eastern side of 
him. To make this 
more evident, let us 
suppose that the small 
circle aon (Fig. 3) rep- 
resents a section of 
the earth perpendic- 
ular to the axis of ro- 
tation of the imagi- 
nary sphere of the 
heavens, (8,*) con- 
ceived to pass through 
the earth's centre ; the 
large circle H Z S a 
section of the heavens 
perpendicular to the 
same line, and pass- 
ing through the sun ; 
and the right line 
H o r the plane of the horizon at the station o. The direction of 
the diurnal motion is from H towards Z and S. Suppose that an 
hour or so after sunset the sun is at S, and that the star r is seen 
in the western horizon; also that the stars t, u, v, &c, are so dis- 
tributed that the distances rt, tu, uv, &c. are each equal to S r. 
Then, at the end of two or three weeks, an hour after sunset the 
star t will be in the horizon ; at the end of another interval of two 
or tliree weeks the star u will be in the same situation at the same 
hour ; at the end of another interval, the star v, &c. It is plain, 
then, that the sun must at the ends of these successive intervals be 
in the successive positions in the heavens, r, t, u, &c. ; otherwise, 
when he is brought by his diurnal motion to the point S, below the 
horizon, the stars t, u, v, &c, could not be successively in the 
plane of the horizon at r. Whence it appears that he has a mo- 
tion in the heavens in the direction S r t u v, opposite to that of 
the diurnal motion ; that is, towards the east 

Another proof of the progressive motion of the sun among the 
stars from west to east, is found in the fact that the same stars rise 
and set earlier each successive night, and week, and month during 
the year. At the end of six months the same stars rise and set 12 
hours earlier ; which shows that the sun accomplishes half a revo- 
lution in this interval. At the end of a year, or of 363 days, the 
stars rise and set again at the same hours, from which it appears 
that the sun completes an entire revolution in the heavens in this 
period of time. 

It is to be observed that the sun does not advance directly to- 
wards the east. He has also some motion from south to north, and 



* Numbers thus enclosed in a parenthesis refer, in general, to a previous article, 



8 INTRODUCTION. 

north to south. It is a matter of common observation that the sun 
is moving towards the north from winter to summer, and towards 
the south from summer to winter. 

17. When the place of the sun in the heavens is accurately 
found from day to day by certain methods of observation, hereaf- 
ter to be explained, it appears that his path is an exact circle, in- 
clined about 23° to a circle running due east and west, (10.) 

18. The motions of the sun, moon, and planets are for the most 
part confined to a certain zone, of about 18° in breadth, extending 
around the heavens from west to east, (or nearly so,) which has 
received the name of the Zodiac. 

19. There is yet another class of bodies, called Comets* (or 
hairy Stars,) that have a motion among the fixed stars. They ap- 
pear only occasionally in the heavens, and continue visible only 
for a few weeks or months. They shine with a diffusive nebu- 
lous light, and are commonly accompanied by a fainter divergent 
stream of similar light, called a tail. 

20. The motions of the comets are not restricted to the zodiac. 
These bodies are seen in all parts of the heavens, and moving in 
every variety of direction. 

21. By inspecting the planets with telescopes, it has been dis- 
covered that some of them are constantly attended by a greater or 
less number of small stars, whose positions are continually vary- 
ing. These attendant stars are called Satellites. The planets 
which have satellites are Jupiter, Saturn, and Uranus. The sat- 
ellites are sometimes called Secondary Planets ; the planets upon 
which they attend being denominated Primary Planets. 

22. The sun and moon, the planets, (including the earth,) to- 
gether with their satellites, and the comets, compose the Solar 
System. 

23. From the consideration of the apparent motions and other 
phenomena of the solar system, several theories have been form- 
ed in relation to the arrangement and actual motions in space of 
the bodies that compose it. The theory, or system, now univer- 
sally received, is (in its most prominent features) that which was 
taught by Copernicus in the sixteenth century, and which is known 
by the name of the Copemican System. It is as follows : 

The sun occupies a fixed centre, about which the planets (in- 
cluding the earth) revolve from west to east,! in planes that are but 
slightly inclined to each other, and in the following order : Mer- 
cury, Venu-s, the Earth, Mars, Vesta, Juno, Ceres, Pallas, Jupiter, 

* From Coma, a head of hair. 

t A motion in space from west to east is a motion from right to left, to a person 
situated within the orbit described, and on the north side of its plane. To obtain 
a clear conception of the motions of the solar system, the reader must place him- 
self, in imagination, in some such situation as this, entirely detached from the 
earth and all the other bodies of the system. It is customary to take the plane of 
the earth's, orbit as the plane of reference in conceiving of the planetary motions. 



GENERAL PHENOMENA OF THE HEAVENS. 9 

Saturn, and Uranus. The earth rotates from west to east, about 
an axis inclined to the plane of its orbit under an angle of about 
66|- , and which remains continually parallel to itself as the earth 
revolves around the sun. The moon revolves from west to east 
around the earth as a centre ; and, in like manner, the satellites 
circulate from west to east around their primaries. Without the 
solar system, and at immense distances from it, are the fixed stars. 
(See the Frontispiece, which is a diagram of the solar system in 
projection.) 

24. We shall here, at the outset, adopt this system as an hypo- 
thesis, and shall rely upon the simple and complete explanations it 
affords of the celestial phenomena, as they come to be investi- 
gated, together with the evidence furnished by Physical Astrono- 
my, to produce entire conviction of its truth in the mind of the 
student. 

25. The following are the characters or symbols employed by 
astronomers for denoting the several planets, and the sun and 
moon : — 

The Sun, © 

Mercury, $ 

Venus, 2 

The Earth, .... 

Mars, <? 

Vesta, fi 

Juno, 5 

26. The two planets, Mercury and Venus, whose orbits lie with- 
in the earth's orbit, are called Inferior Planets. The others are 
called Superior Planets. 

27. The angular distance between any two fixed stars is found 
to be the same, from whatever point on the earth's surface it is 
measured. It follows, therefore, that the diameter of the earth is 
insensible, when compared with the distance of the fixed stars ; 
and that, with respect to the region of space which separates us 
from these bodies, the whole earth is a mere point. Moreover, the 
angular distance between any two fixed stars is the same at what- 
ever period of the year it is measured. Whence, if the earth re- 
volves around the sun, its entire orbit must be insensible in com- 
parison with the distance of the stars. 

28. On the hypothesis of the earth's rotation, the diurnal motion 
of the heavens is a mere illusion, occasioned by the rotation of the 
earth. To explain this, suppose the axis of the earth prolonged on 
till it intersects the heavens, considered as concentric with the 
earth. Conceive a great circle to be traced through the two points 
of intersection and the point directly over head, and let the position 
of the stars be referred to this circle. It will be readily perceived 
that the relative motion of this circle and the stars will be the same, 
whether the circle rotates with the earth from west to east, or, the 

2 



Ceres, . . . 


. ? 


Pallas, . . . 


. $ 


Jupiter, . . . 


. u 


Saturn, . . . 


. T? 


Uranus, . . 


. W 


The Moon, 


. D 



10 



INTRODUCTION. 



earth being stationary, the whole heavens rotate about the same 
axis and at the same rate in the opposite direction. Now, as the 
motion of the earth is perfectly equable, we are insensible of it, 
and therefore attribute the changes in the situations of the stars 
with respect to the earth to an actual motion of these bodies. It 
follows, then, that we must coik eive the heavens to rotate as above 
mentioned, since, as we have seen, such a motion would give rise 
to the same changes of situation as the supposed rotation of the 
earth. It was stated (Art. 8) that the sphere of the heavens ap- 
pears to rotate about a line passing through the north pole and the 
station of the observer ; but, as the radius of the earth is insensi- 
ble in comparison with the distance of the stars, an axis passing 
through the centre of the earth will, in appearance, pass through 
the station of the observer, wherever this may be upon the earth's 
surface. 

29. We in like manner infer that the observed motion of the 
sun in the heavens is only an apparent motion, occasioned by the 



Fig. 4. 



orbitual motion of the earth. 
Let E, E' (Fig. 4) represent 
two positions of the earth in 
its orbit EE'E" about the 
sun S. When the earth is 
at E, the observer will refer 
the sun to that part of the 
heavens marked s; but when 
the earth is arrived at E', he 
will refer it to the part mark- 
ed s' ; and being in the mean 
time insensible of his own 
motion, the sun will appear 
to him to have described in 
the heavens the arc 5 s', just 
the same as if it had actu- 
ally passed over the arc SS' 
m space, and the earth had, during that time, remained quiescent 
at E. The motion of the sun from s towards s f will be from west 
to east, since the motion of the earth from E towards E' is in this 
direction. Moreover, as the axis of the earth is inclined to the 
plane of its orbit under an angle of 66%°. (23,) the plane of the 
sun's apparent path, which is the same as that of the earth's orbit, 
will be inclined 23 \° to a circle perpendicular to the earth's axis, 
or to a circle directed due east and west. 




PART I. 

ON THE DETERMINATION OF THE PLAGES AND MOTIONS 
OF THE HEAVENLY BODIES. 



CHAPTER I. 

ON THE CELESTIAL AND TERRESTRIAL SPHERES. 

30. In determining from observation the apparent positions and 
motions of the heavenly bodies, and, in general, in all investigations 
that have relation to their apparent positions and motions, Astron- 
omers conceive all these bodies, whatever may be their actual 
distance from the earth, to be referred to a spherical surface of an 
indefinitely great radius, having the station of the observer, or 
what comes to the very same thing, the centre of the earth, for its 
centre. This imaginary spherical surface is called the Sphere of 
the Heavens, or the Celestial Sphere. It is important to observe, 
that by reason of the great dimensions of this sphere, if two lines 
be drawn through any two points of the earth, and parallel to each 
other, they will, when indefinitely prolonged, meet it sensibly in 
the same point ; and that, if two parallel planes be passed through 
any two points of the earth, they will intersect it sensibly in the 
same great circle. This amounts to saying that the earth, as com- 
pared to this sphere, is to be considered as a mere point at its 
centre. 

31. Not only is the size of the earth to be neglected in compari- 
son with the celestial sphere, but also the size of the earth's orbit. 
Thus the supposed annual motion of the earth around the sun, 
does not change the point in which a line conceived to pass from 
any station upon the earth in any fixed direction into space, pierces 
the sphere of the heavens ; nor the circle in which a plane cuts the 
same sphere. 

The fixed stars are so remote from the earth that observers, 
wherever situated upon the earth, and in the different positions of 
the earth in its orbit, refer them to the same points of the celestial 
sphere, (27.) The other heavenly bodies are referred by observ- 
ers at different stations to points somewhat different. 

32. For the purposes of observation and computation, certain 
imaginary points, lines, and circles, appertaining to the celestial 
sphere, are employed, which we shall now proceed to explain. 

(1.) The Vertical Line, at any place on the earth's surface, is 



12 



ON THE CELESTIAL SPHERE. 



the line of descent of a falling body, or the position assumed by a 
plumb-line when the plummet is freely suspended and at rest. 

Every plane that passes through the vertical line is called a Ver- 
tical Plane. Every plane that is perpendicular to the vertical line, 
is called a Horizontal Plane. 

(2.) The Sensible Horizon of a place on the earth's surface, is 
the circle in which a horizontal plane, drawn through the place, 
cuts the celestial sphere. As its plane is tangent to the earth*, it 
separates the visible from the invisible portion of the heavens, (5.) 
(3.) The Rational Horizon is a circle parallel to the former, 
the plane of which passes through the centre of the earth. The 
zone of the heavens comprehended between the sensible and ra- 
tional horizon is imperceptible, or the two circles appear as one 
and the same at the distance of the earth, (30.) 

(4.) The Zenith of a place is the point in which the vertical 
prolonged upward pierces the celestial sphere. The point in 
which the vertical, when produced downward, intersects the ce- 
lestial sphere, is called the Nadir. 

The zenith and nadir are the geometrical poles of the horizon. 
(5.) The Axis of the Heavens is an imaginary right line pass- 
ing through the north pole (8) and the centre of the earth. It is 
the line about which the apparent rotation of the heavens is per- 
formed. It is, also, on the hypothesis of the earth's rotation, the 
axis of rotation of the earth prolonged on to the heavens. 

(6.) The South Pole of the heavens is the point in which the 
axis of the heavens meets the southern part of the celestial sphere. 

To illustrate the 
preceding definitions, 
let the inner circle 
n s (Fig. 5) repre- 
sent the earth, and the 
outer circle HZRN 
the sphere of the 
heavens ; also let 
be a point on the 
earth's surface, and 
OZ the vertical line 
at the station O. — 
Then HOR will be 
the plane of the sen- 
sible horizon, HCR 
the plane of the ra- 
tional horizon, Z the 
zenith, and N the na- 
dir ; and if P be the 
north pole of the hea- 
vens, OP, or CP its parallel, will be the axis of the heavens, and 
P' the south pole. 




DEFINITIONS 



13 



The lines CP and OP intersect the heavens in the same point, 
P; and the planes HOR, and HCR, in the same circle, passing 
through the points H and R. 

Unless we are seeking for the exact apparent place in the heav- 
ens of some other heavenly body than a fixed star, we may con- 
ceive the observer to be stationed at the earth's centre, in which 
case OP will become the same 
as *CP, and HOR the same- as 
HCR ; as represented in Fig. 6. 
[n this diagram, the circle of the 
horizon being supposed to be view- 
ed from a point above its plane, is 
represented by the ellipse HARa. 
Z and N are its geometrical poles, h^ 
In the construction of Fig. 5 the 
eye is supposed to be in the plane 
of the horizon, and HARa is pro- 
jected into its diameter HCR. 

Every different place on the 
surface of the earth has a different 
zenith, and, except in the case of diametrically opposite places, a 
different horizon. To illustrate this, let nesq (Fig. 7) represent 
the earth, and HZRP' the sphere of the heavens ; then, considering 
the four stations, e, 0, n, and q, the zenith and horizon of the first 




Fig. 7. 



will be respectively E 
and PeP' ; of the se- 
cond Z and HOR; of 
the third P and QnE ; 
of the fourth Q and 
P'?P. The diametri- 
cally opposite places 
O and O' have the 
same rational horizon, 
viz. HCR. The same 
is true of the places n 
and s, and e and q. 
Their rational hori- 
zons are respectively 
QCE and PCP'. 

(7.) Vertical Circles 
are great circles pass- 
ing through the zenith 
and nadir. They cut 
the horizon at right angles, and their planes are vertical. Thus, 
ZSM (Fig.6) represents a vertical circle passing through the stai 
S, called the Vertical Circle of the Star. 

(8.) The Meridian of a place is that vertical circle which con- 




14 ON THE CELESTIAL SPHERE. 

tains the north and south poles of the heavens. The plane of the 
meridian is called the Meridian Plane. 

Thus, PZRP' is the meridian of the station C. The half 
HZR, above the horizon, is termed the Superior Meridian, and 
the other half RNH, below the horizon, is termed the Inferior 
Meridian. The two points, as H and R, in which the meridian 
cuts the horizon, are called the North and South Points of the 
horizon ; and the line of intersection, as HCR, of the meridian 
plaLe with the plane of the horizon, is called the Meridian Line, 
or the North and South Line. 

(9.) The Prime Vertical is the vertical circle which crosses the 
meridian at right angles. It cuts the horizon in two points, as 
e, w, called the East and West Points of the Horizon. 

(10.) The Altitude of any heavenly body is the arc of a vertical 
circle, intercepted between the centre of the body and the horizon, 
or the angle at the centre of the sphere, measured by this arc. 
Thus, SM or MCS is the altitude of the star S. 

(11.) The Zenith Distance of a heavenly body is the arc of a 
vertical circle, intercepted between its centre and the zenith; or 
the distance of the centre of the body from the zenith, as meas- 
ured by the arc of a great circle. Thus, ZS, or ZCS, is the 
zenith distance of the star S. 

It is obvious that the zenith distance and altitude of a body are 
complements of each other, and therefore when either one is known 
that the other may be found. 

(12.) The Azimuth of a heavenly body is the arc of the horizon, 
intercepted between the meridian and the vertical circle passing 
through the centre of the body ; or the angle comprehended be- 
tween the meridian plane and the vertical plane containing the 
centre of the body. It is reckoned either from the north or from 
the south point, and each way from the meridian. HM or HCM 
represents the azimuth of the star S. 

The Azimuth and Altitude, or azimuth and zenith distance of 
a heavenly body, ascertain its position with respect to the horizon 
and meridian, and therefore its place in the visible hemisphere. 
Thus, the azimuth HM determines the position of the vertical cir- 
cle ZSM of the star S with respect to the meridian ZPH, and the 
altitude MS, or the zenith distance ZS, the position of the star in 
this circle. 

(13.) The Amplitude of a heavenly body at its rising, is the arc 
of the horizon intercepted between the point where the body rises 
and the east point. Its amplitude at setting, is the arc of the ho- 
rizon intercepted between the point where the body sets and the 
west point. It is reckoned towards the north, or towards the south, 
according as the point of rising or setting is north or south of the 
east or west point. Thus, if aBSA represents the circle described 
by the star S in its diurnal motion, ea will be its amplitude at 
rising, and wA. its amplitude at setting. 



DEFINITIONS 15 

(14.) The Celestial Equator, or the Equinoctial, is a great cir- 
cle of the celestial sphere, the plane of which is perpendicular to 
the axis of the heavens. The north and south poles of the heav- 
ens are therefore its geometrical poles. The celestial equator is 
represented in Fig. 6 by EwQe. This circle is also frequently 
called the Equator, simply. 

(15.) Parallels of Declination are small circles parallel to the 
celestial equator. cBSA represents the parallel of declination 
of the star S. 

The parallels of declination passing through the stars, are the 
circles described by the stars in their apparent diurnal motion. 
These, by way of abbreviation, we shall call Diurnal Circles. 

(16.) Celestial Meridians, Hour Circles, and Declination Cir- 
cles, are different names given to all great circles which pass 
through the poles of the heavens, cutting the equator at right an- 
gles. PSP' is a celestial meridian. The angles comprehended 
between the hour circles and the meridian, reckoning from the 
meridian towards the west, are called Hour Angles, or Horary 
Angles. 

(17.) The Ecliptic is that great circle of the heavens which the 
sun appears to describe in the course of the year. 

(18.) The Obliquity of the Ecliptic is the angle under which 
the ecliptic is inclined to the equator. Its amount is 23 \°. 

(19.) The Equinoctial Points are the two points in which the 
ecliptic intersects the equator. That one of these points which the 
sun passes in the spring is called the Vernal Equinox, and the 
other, which is passed in the autumn, is called the Autumnal Equi- 
nox. These terms are also applied to the epochs when the sun is 
at the one or the other of these points. These epochs are, for the 
vernal equinox the 21st of March, and for the autumnal equinox 
the 23d of September, or thereabouts. 

(20.) The Solstitial Points are the two points of the ecliptic 
90° distant from the vernal and autumnal equinox. The one that 
lies to the north of the equator is called the Summer Solstice, and 
the other the Winter Solstice. The epochs of the sun's arrival 
at these points are also designated by the same terms. The sum- 
mer solstice happens about the 21st of June, and the winter solstice 
about the 22d of December. 

(21.) The Equinoctial Colure is the celestial meridian passing 
through the equinoctial points ; and the Solstitial Colure is the ce- 
lestial meridian passing through the solstitial points. 

(22.) The Polar Circles are parallels of declination at a distance 
from the poles equal to the obliquity of the ecliptic. The one 
about the north pole is called the Arctic Circle ; the other, about 
the south pole, is called the Antarctic Circle. 

The polar circles contain the geometrical poles of the ecliptic. 

(23.) The Tropics are parallels of declination at a distance from 
the equator equal to the obliquity of the ecliptic. That which is 



16 



OX THE CELESTIAL SPHERE. 



on the north side of the equator is called the Tropic of Cancer, 
and the other the Tropic of Capricorn. 

The tropics touch the ecliptic at the solstitial points. 




Let C (Fig. 8) represent the centre of the earth and sphere, 
PCP' the axis of the heavens, EVQA the equator, WVTA the 
ecliptic, and K, K', its poles. Then will V be the vernal and A 
the autumnal equinox ; W the winter, and T the summer solstice ; 
PVP'A the equinoctial cohere ; PKWK'T the solstitial colure ; 
the angle TCQ, or its measure the arc TQ, the obliquity of the 
ecliptic; K;;iU, K'ra'U', the polar circles ; and TnZ, Wn'Z', the 
tropics. 

It is important to observe that, agreeably to what has been sta- 
ted, (Art. 30,) the directions of the equator and ecliptic, of the equi- 
noctial points, and of the other points and circles just defined and 
illustrated, are the same at any station upon the surface of the 
earth as at its centre. Thus, the equator lies always in the plane 
passing through the place of observation, wherever this may be, 
and parallel to the plane which, passing through the earth's centre, 
cuts the heavens in this circle. In like manner the echptic lies, 
everywhere, in a plane parallel to that which is conceived to pass 
through the centre of the earth and cut the heavens in this circle, 
and so for the other circles. 

(24.) The Zodiac (18) extends about 9° on each side of the 
ecliptic. 



DEFINITIONS 17 

(25.) The ecliptic and zodiac are divided into twelve equal parts, 
called Signs. Each sign contains 30°. The division commences 
at the vernal equinox. Setting out from this point, and following 
around from west to east, the Signs of the Zodiac, with the re- 
spective characters by which they are designated, are as follows : 
Aries T, Taurus S, Gemini II, Cancer S>, Leo SI, Virgo ty, Li- 
bra =£=, Scorpio fTj,, Sagittarius X , Capricornus V3, Aquarius zz, 
Pisces }£. The first six are called northern signs, being north of 
the equinoctial. The others are called southern signs. 

The vernal equinox corresponds to the first point of Aries, and 
the autumna'l equinox to the first point of Libra. The summer 
solstice corresponds to the first point of Cancer, and the winter 
solstice to the first point of Capricornus. 

The mode of reckoning arcs on the ecliptic is by signs, degrees, 
minutes, &c. 

A motion in the heavens in the order of the signs, or from' west 
to east, is called a direct motion, and a motion contrary to the or- 
der of the signs, or from east to west, is called a retrograde mo- 
tion. 

(26.) The Right Ascension of a heavenly body is the arc of the 
equator intercepted between the vernal equinox and the declination 
circle which passes through the centre of the body, as reckoned 
from the vernal equinox towards the east. It measures the incli- 
nation of the declination circle of the body to the equinoctial colure. 
Thus, PSR being the declination circle of the star S, and V the 
place of the vernal equinox, VR is the right ascension of the star.. 
It is the measure of the angle VPS. If PS'R' be the declination 
circle of another star S', SPS', or RR', will be their difference of 
right ascension. 

(27.) The Declination of a heavenly body is the arc of a circle 
of declination, intercepted between the centre of the body and the 
equator. It therefore expresses the distance of the body from the 
equator. Thus, RS is the declination of the star S. 

Declination is North or South, according as the body is north or 
south of the equator. 

In the use of formulae, a south declination is regarded as nega- 
tive. 

The right ascension and declination of a heavenly body are two 
co-ordinates, which, taken together, fix its position in the sphere 
of the heavens : for they make known its situation with respect to 
two circles, the equinoctial colure, and the equator. Thus, VR 
and RS ascertain the position of the star S with respect to the cir- 
cles PVP'A, and VQAE. 

(28.) The Polar Distance of a heavenly body is the arc of a de- 
clination circle, intercepted between the centre of the body and the 
elevated pole. The polar distance is the complement of the decli- 
nation, and, therefore, when either is known the other may be 
found. 

3 



18 ON THE TERRESTRIAL SPHERE. 

(29.) Circles of Latitude are great circles of the celestial sphere, 
which pass through the poles of the ecliptic, and therefore cut this 
circle at right angles. Thus, KSL represents a part of the circle 
of latitude of the star S. 

(30.) The Longitude of a heavenly body is the arc of the eclip- 
tic, intercepted between the vernal equinox and the circle of lati- 
tude which passes through the centre of the body, as reckoned 
from the vernal equinox towards the east, or in the order of the 
signs. It measures the inclination of the circle of latitude of the 
body to the circle of latitude passing through the vernal equinox. 
Thus, VL is the longitude of the star S. It is the measure of the 
angle VKS. 

(31.) The Latitude of a heavenly body is the arc of a circle of 
latitude, intercepted between the centre of the body and the eclip- 
tic. It therefore expresses the distance of the body from the eclip- 
tic. Thus, LS is the latitude of the star S. 

Latitude is North or South, according as the body is north or 
south of the ecliptic. 

In the use of formulae, a south latitude is affected with the mi- 
nus sign. 

The longitude and latitude of a heavenly body are another set 
<of co-ordinates, which serve to fix its position in the heavens. They 
ascertain its situation with respect to the circle of latitude passing 
through the vernal equinox and the ecliptic. Thus, VL and LS 
fix the position of the star S, making known its situation with re- 
spect to the circles KVK'A and VTAW. 

(32.) The Angle of Position of a star, is the angle included at 
the star between the circles of latitude and declination passing 
through it. PSK is the angle of position of the star S. 

(33.) The Astronomical Latitude, or the Latitude, of a place, is 
the arc of the meridian intercepted between the zenith and the ce- 
lestial equator. It is Noi'th or South, according as the zenith is 
north or south of the equator. ZE (Fig. 7) represents the latitude 
of the station O ; QOE or QCE being the equator. 

33. The earth's surface, considered as spherical, (which ac- 
curate admeasurement, upon principles that will be explained in 
the sequel, proves it to be, very nearly,) is called the Terrestrial 
Sphere. The following geometrical constructions appertain to the 
terrestrial sphere, as it is employed for the purposes of astronomy. 
It will be observed that they correspond to those of the celestial 
sphere above described, and are used for similar objects. 

(1.) The North and South Poles of the Earth are the two points 
in which the axis of the heavens intersects the terrestrial sphere. 
They are also the extremities of the earth's axis of rotation. 

(2.) The Terrestrial Equator is the great circle in which a 
plane passing through the centre of the earth, and perpendicular to 
the axis of the heavens and earth, cuts the terrestrial sphere. The 
terrestrial and the celestial equator, then, lie in the same plane. 



DEFINITIONS 



19 



The poles of the earth are the geometrical poles of the terrestrial 
equator. The two hemispheres into which the terrestrial equator 
divides the earth, are called, respectively, the Northern Hemi- 
sphere and the Southern Hemisphere. 

(3.) Terrestrial Meridians are great circles of the terrestrial 
sphere, passing through the north and south poles of the earth, and 
cutting the equator at right angles. Every plane that passes through 
the axis of the heavens, cuts the celestial sphere in a celestial me- 
ridian, and the terrestrial sphere in a terrestrial meridian. 

Let PP' (Fig. 9) represent the axis of the heavens, the centre 
of the earth, andj? andp' its poles. Then, elq will represent the 

Fig. 9. 




terrestrial equator, (ELQ representing the celestial equator;) and 
pep' zndpsp' terrestrial meridians, (PEP' and PSP' representing 
celestial meridians.) 

(4.) The Reduced Latitude of a place on the earth's surface is 
the arc of the terrestrial meridian, intercepted between the place 
and the equator, or the angle at the centre of the earth measured 
by this arc. Thus, oe, or the angle oOe, is the reduced latitude 
of the place o. Latitude is North or South, according as the 
place is north or south of the equator. The reduced latitude dif- 
fers somewhat from the astronomical latitude, by reason of the 
slight deviation of the earth from a spherical form. Their differ- 
ence is called the Reduction of Latitude. 

(5.) Parallels of Latitude are small circles of the terrestrial 



20 



ON THE TERRESTRIAL SPHERE. 



sphere parallel to the equator. Every point of a parallel of latitude 
has the same latitude. 

The parallels of latitude which correspond in situation with the 
polar circles and tropics in the heavens, have received the same 
appellations as these circles. (See defs. 22, 23, p. 15.) 

(6.) The Longitude of a place on the earth's surface, is the in- 
clination of its meridian to that of some particular station, fixed 
upon as a circle to reckon from, and called the First Meridian. It 
is measured by the arc of the equator, intercepted between the first 
meridian and the meridian passing through the place, and is called 
East or West, according as the latter meridian is to the east or to 
the west of the first meridian. Thus, if pqp' be supposed to re- 
present the first meridian, the angle spq, or the arc ql, will be the 
longitude of the place s. 

Different nations have, for the most part, adopted different first 
meridians. The English use the meridian which passes through 
the Royal Observatory at Greenwich, near London ; and the 
French, the meridian of the Royal Observatory at Paris. In the 
United States the longitude is, for astronomical purposes, reckoned 
from the meridian of Greenwich or Paris, (generally the former.) 

The longitude and latitude of a place designate its situation on 
the earth's surface. They are precisely analogous to the right as- 
cension and declination of a star in the heavens. 

34. The diagram (see Fig. 6) which we made use of in Art. 32 
in illustrating our description of the circles of the celestial sphere,, 
represents the aspect of this sphere at a place at which the north 



Fig. 10. 



pole of the heavens is some- 
where between the zenith and 
horizon. Such is the position 
of the north pole at all places 
situated between the equator 
and the north pole of the 
earth. For, let O (Fig. 10) 
represent a place on the earth's 
surface, HOR the horizon,, 
OZ the vertical, HZR the 
meridian, and ZE the latitude, 
QOE will then represent the 
equinoctial, andP,P', 90° dis- 
tant from E and on the meri- 
dian, the poles . Now, we have 
HP = ZH — ZP = 90° ~ ZP ; ZE = PE — ZP = 90° — ZP„ 
Whence HP = ZE. 
Thus, the altitude of the pole is everywhere equal to the latitude 
of the place. It follows, therefore, that in proceeding from the 
equator to the north pole, the altitude of the north pole of the heav- 
ens will gradually increase from 0° to 90°. 

By inspecting Fig. 7, it will be seen that this increase of the al- 




ASPECTS OF THE CELESTIAL SPHERE. 



21 



titude of the pole in going north, is owing to the fact that in fol- 
lowing the curved surface of the earth the horizon, which is con- 
tinually tangent to the earth, is being constantly more and more 
depressed towards the north, while the absolute direction of the 
pole remains unaltered. 

If the spectator is in the southern hemisphere, the elevated pole, 
as it is always on the opposite side of the zenith from the equator, 
will be the south pole. At corresponding situations of the spec- 
tator it will obviously have the same altitude as the north pole in 
the northern hemisphere. 

35. Let us now inquire minutely into the principal circumstan- 
ces of the diurnal motion of the stars, as it is seen by a spectator 
situated somewhere between the equator and the north pole. And 
in the first place, it is a simple corollary from the proposition just 
established, that the parallel of declination to the north, whose 
polar distance is equal to the latitude of the place, will lie entirely 
above the horizon, and just touch it at the north point. This cir- 
cle is called the circle of perpetu- Fig. 11. 
al apparition; the line <zH(Fig. 11) 
represents its projection on the me- 
ridian plane. The stars compre- 
hended between it and the north 
pole will never set. As the de- 
pression of the south pole is equal 
to the altitude of the north pole, h 
the parallel of declination o R, at 
■a distance from the south pole 
equal to the latitude of the place, 
will lie entirely below the horizon, 
and just touch it at the south point 
The parallel thus situated is call- 
ed the circle of perpetual occupation. The stars comprehended 
between it and the south pole will never rise. 

The celestial equator (which passes through the east and west 
points) will intersect the meridian at a point E, whose zenith dis- 
tance ZE is equal to the latitude of the place (Def. 33, Art. 32,) and 
consequently, whose altitude RE is equal to the co-latitude of the 
place. Therefore, in the situation of the observer above supposed, 
the equator QOE, passing to the south of the zenith, will, togeth- 
er with the diurnal circles nr, st, &a, which are all parallel to it, 
be obliquely inclined to the horizon, making with it an angle equal 
to the co-latitude of the place. As the centres c,c', &c, of the 
diurnal circles he on the axis of the heavens, which is inclined to 
the horizon, all diurnal circles situated between the two circles of 
perpetual apparition and occultation, aR and oR, with the excep- 
tion of the equator, will be divided unequally by the horizon. The 
greater parts of the circles nr, n'r', &c., to the north of the equa- 
tor, will be above the horizon : and the greater parts of the circles 




Wt^ffl 



22 ON THE CELESTIAL SPHERE. 

st, s't', &c., to the south of the equator, will be below the horizon 
Therefore, while the stars situated in the equator will remain an 
equal length of time above and below the horizon, those to the. 
north of the equator will remain a longer time above the horizon 
than below it ; and those to the south of the equator, on the con- 
trary, a longer time below the horizon than above it. It is also 
obvious, from the manner in which the horizon cuts the different 
diurnal circles, that the disparity between the intervals of time that 
a star remains above and below the horizon, will be the greater the 
more distant it is from the equator. Again, the stars will all cul- 
minate, or attain to their greatest altitude, in the meridian : for, 
since the meridian crosses the diurnal circles at right angles, they 
will have the least zenith distance when in this circle.. Moreover, 
as the meridian bisects the portions of the diurnal circles which lie 
above the horizon, the stars will all employ the same length of 
time in passing from the eastern horizon to the meridian, as in 
passing from the meridian to the western horizon. The circum- 
polar stars will pass the meridian twice in 24 hours ; once above, 
and once below the pole. These meridian passages are called, 
respectively, Upper and Lower Culminations, or Inferior and Su- 
perior Transits. 

It will be observed, that in travelling towards the north the cir- 
cles of perpetual apparition and occupation, together with those 
portions of the heavens about the poles which are constantly visible 
and invisible, are continually on the increase. 

It is evident, from what is stated in Art. 34, that the circum- 
stances of the diurnal motion will be the same at any place in the 
southern hemisphere, as at the place which has the same latitude 
in the northern. 

The celestial sphere in the position relative to the horizon which 
we have now been considering, which obtains at all places situated 
between the equator and either pole, is called an Oblique Sphere, 
because all bodies rise and set obliquely to the horizon. 

Fig. 12. 36. When the spectator is sit- 

uated on the equator, both the 
celestial poles will be in his hori- 
zon, (34,) and therefore the celes- 
tial equator and the diurnal circles 
in general will be perpendicular to 
the horizon. This situation of the 
sphere is called a Right Sphere, 
for the reason that all bodies rise 
and set at right angles with the 
horizon. It is represented in Fig. 
12. As the diurnal circles are 
bisected by the horizon, the stars 
will all remain the same length of 
time above as below the horizon. 




ASTRONOMICAL INSTRUMENTS. 



23 



37. If the observer be at either 
oi the poles, the elevated pole of 
the heavens will be in his zenith, 
(34,) and consequently, the celes- 
tial equator will be in his horizon. 
The stars will move in circles 
parallel to the horizon, and the 
whole hemisphere, on the side of 
the elevated pole, will be continu- 
ally visible, while the other hem- 
isphere will be continually invis- 
ible. This is called a Parallel 
Sphere. It is represented in 
Fig. 13. 



Fig. 13. 




CHAPTER II. 

ON THE CONSTRUCTION AND USE OF THE PRINCIPAL ASTRONOMICAL 
INSTRUMENTS. 



38. Astronomical Instruments are, for the most part, used for 
the admeasurement of arcs of the celestial sphere, or of angles cor- 
responding to such arcs at the earth's surface. They consist, es- 
sentially, of a refracting telescope turning upon a horizontal axis, 
and of a vertical graduated limb, (or, in some cases, of both a ver- 
tical and a horizontal graduated limb,) to indicate the angle passed 
over by the telescope. At the common focus of the object-glass 
and eye-glass of the telescope is a diaphragm, or circular plate, at- 
tached to which are two very fine wires, or spider-lines, crossing 
each other at right angles in its centre. The place of this dia- 
phragm may be altered by adjusting screws ; it is by this means 
Drought into such a position that the cross of the wires will lie on 
the axis of the telescope, (that is, the line joining the centres of the 
object-glass and eye-glass.) The line joining the centre of the ob- 
ject-glass and the cross of the wires, is technically termed the Line 
of Collimation. Bringing the cross of the wires upon the axis of 
the telescope, is called Adjusting the Line of Collimation. A star 
is known to be On the line of collimation when it is bisected by the 



cross-wires. 



The telescope either turns around the centre of the graduated 
limb, or, which is more common, the limb and telescope are firmly 
attached to each other, and turn together. In the first arrange- 
ment a small steel plate, firmly connected with the telescope, slides 
along the limb. Upon this plate a small mark is drawn, which is 
called the Index. The required angle is read off by noting the 



24 



ASTRONOMICAL INSTRUMENTS. 



angle upon the limb which is pointed out by the index ; the zero 
on the hmb being generally, in practice, the point from which the 
angle is reckoned. When the telescope and graduated limb are 
firmly connected, the limb slides past the index, which is now sta- 
tionary. The limbs of even the largest instruments are not divided 
into smaller parts than about 5', but, by means of certain subsidi- 
ary contrivances, the angle may, with some instruments, be read 
oft to within a fraction of a second. 

39. The principal contrivances for increasing the accuracy of 
the reading off of angles, are the Vernier, the Micrometer Screw, and 
the Micrometer Microscope or Reading Microscope. The Vernier 
is only the index plate, so graduated that a certain number of its 
divisions occupy the same space as a number one less on the limb. 
Fig. 14 represents a vernier and a portion of the limb of the instru- 
ment, 15 divisions on the vernier corresponding to 14 on the limb. 
If we suppose the smallest divisions oAhe limb to be 15', and call 
x the number of minutes in one division of the vernier, then, 
15 x = 14 x 15 ; , and x = 14'. 

Thus, the difference between a division on the vernier and one 
on the limb, will be 1'. Accordingly, if the index, which is the 
first mark on the vernier, should be little past the mark 40° on the 
limb, and the second mark of the vernier should coincide with the 
next point of division, marked 15', the angle would be 40° 1'. If 
the third mark on the vernier were coincident with the next division 
of the limb, marked 30', the angle would be 40° 2'. If the fourth 
with the next division to this, 40° 3' ; and so on. 

By making the divisions on the vernier more numerous, the an- 



Fig. 14. 



gle can be read off with greater 
precision ; but a better expedi- 
ent is provided in the Microme- 
ter Screw. This piece of me- 
chanism is represented in Fig. 
14. The part E can be fast- 
ened to the limb of the instru- 
ment by means of a screw. FG 
is a screw, with a milled head at 
F, working in a collar fixed in 
the under part of E , and in a nut 
fixed in the under part of the tel- 
escope T t. When the part E 
is fixed or clamped, and the 
screw is turned around by its 
milled head at F, it must com- 
municate a direct motion to the 
nut, and, consequently, to the 
telescope and vernier in the direction of FG. Attached to the screw, 
or to the small cylinder on which it is formed, is an index D, move- 
able together with the screw, and on a thin graduated immoveable 




READING MICROSCOPE — TIME, ETC. 25 

plate, the profile only of which is shown in the figure. Suppose 
now that the screw is of such fineness that while, together with the 
index D, it makes a complete revolution, the vernier moves through 
an arc of 1'. Then, if the plate be divided into 60 equal parts, a 
motion of the index over one of these parts would answer to a mo- 
tion of 1" on the limb. This being understood, to show the use of 
the micrometer screw, suppose that no two marks on the vernier 
and limb are coincident : bring the two nearest into coincidence by 
turning the screw, and the number of divisions passed over by the 
index D will be the seconds to be added to or subtracted from the 
angle read off with the vernier. In observing the coincidence of 
the divisions of the limb and vernier, the eye is assisted by a mi- 
croscope.* 

40. The Reading Microscope is a compound microscope firmly 
fixed opposite to the limb, and furnished with cross-wires in the focus 
of the eye-glass, or conjugate focus of the object-glass, moveable by 
a fine-threaded micrometer screw, that is, a screw (such as was de- 
scribed in the previous article) provided with an immoveable grad- 
uated circular plate, and an index turning with the screw, and glid- 
ing over the plate, to measure the exact distance through which the 
head of the screw is moved. The observer looks through the 
microscope at the limb. The centre of the microscope corresponds 
to the index of a fixed vernier plate. By turning the screw the 
intersection of the wires is moved over the space which separates 
it from the nearest line of division on the limb, in the direction of 
the zero, and the number of turns and parts of a turn of the screw 
being noted by means of the graduated plate, the number of mi- 
nutes and seconds in this space becomes known. The minutes 
and seconds thus found being added to the angle read off from the 
limb, the result will be the angle sought. 

41. It is obvious that, other things being the same, instruments 
are accurate in proportion to the power of the telescope and the 
size of the limb. The large instruments now in use in astronomi- 
cal observatories, are relied upon as furnishing angles to within 1" 
of the truth. 

42. Time is an essential element in astronomical observation. 
Three different kinds of time are employed by astronomers : Si- 
dereal, Apparent or True Solar, and Mean Solar Time. 

43. Sidereal Time is time as measured by the diurnal motion 
of the stars, or, more properly, of the vernal equinox. A Sidereal 
Day is the interval between two successive meridian transits of a 
star, or, (as it is now most generally considered,) the interval be- 
tween two successive transits of the vernal equinox. It commences 
at the instant when the vernal equinox is on the superior meridian, 
and is divided into 24 Sidereal Hours. 

44. Apparent, or True Solar Time, is deduced from observa- 

* Woodhouse's Astronomy, vol i. p. 55. 
4 



26 ASTRONOMICAL INSTRUMENTS. 

tions upon the sun. An Apparent Solar Day is the interval be- 
tween two successive meridian passages of the sun's centre ; com- 
mencing when the sun is on the superior meridian. It appears 
from observation that it is a little longer than a sidereal day, and 
that its length is variable during the year. It is divided into 24 
Apparent Solar Hours. 

45. Mean Solar Time is measured by the diurnal motion of an 
imaginary sun, called the Mean Sun, conceived to move uniformly 
from west to east in fhe equator, with the real sun's mean motion 
in the ecliptic, and to have at all times a right ascension equal to 
the sun's mean longitude. A Mean Solar Day commences when 
the mean sun is on the superior meridian, and is divided into 24 
Mean Solar Hours. 

Since the mean sun moves uniformly and directly towards the 
east, the length of the mean solar day must be invariable. 

46. The Astronomical Day commences at noon, and is divided 
into 24 hours ; but the Calendar Day commences at midnight, and 
is divided into two portions of 12 hours each. 

47. Astronomical observations are, for the most part, made in 
the plane of the meridian. But some of minor importance are 
made out of this plane. The chief instruments employed for me- 
ridian observations, are the Astronomical Circle, and the Transit 
Instrument, used in connection with the Astronomical Clock. 
These are the capital instruments of an observatory, inasmuch as 
they serve (as will soon be explained) for the determination of the 
places of the heavenly bodies, which are the fundamental data of 
astronomical science. The principal instruments used for making 
observations out of the meridian plane, are the Altitude and Azi- 
muth Instrument, the Equatorial, and the Sextant. 

TRANSIT INSTRUMENT. 

48. The Transit Instrument is a meridional instrument, employ- 
ed in conjunction with a clock or chronometer for observing the 
passage of celestial objects across the meridian, either for the pur- 
pose of determining their difference of right ascension, or obtaining 
the correct time. It is constructed of various dimensions, from a 
focal length of 20 inches to one of 10 feet. The larger and more 
perfect instruments are permanently fixed in the meridian plane ; 
the smaller ones are mounted upon portable stands. Fig. 15 rep- 
resents a fixed transit instrument. AD is a telescope, fixed, as it 
is represented in the figure, to a horizontal axis formed of two 
cones. The two small ends of these cones are ground into two 
perfectly equal cylinders ; which cylindrical ends are called Pivots. 
These pivots rest on two angular bearings, in form like the upper 
part of a Y, and denominated Y's. The Y's are placed in two 
dove-tailed brass grooves fastened in two stone pillars E and W, so 
erected as to be perfectly steady. One of the grooves is horizontal, 
the other vertical, so that, by means of screws, one end of the axis 



TRANSIT INSTRUMENT. 27 

may be pushed a little forward or backward, and the other end 
may be either slightly depressed or elevated : which two small 
movements are necessary, as it will be soon explained, for two ad 
justments of the telescope. 

Fig. 15. 




Let E be called the eastern pillar, W the western. On the 
eastern end of the axis is fixed (so that it revolves with the axis) 
an index n, the upper part of which, when the telescope revolves, 
nearly slides along the graduated face of a circle, attached, as it is 
shown in the figure, to the eastern pillar. The use of this part of 
the apparatus is to adjust the telescope to the altitude or zenith 
distance of a star the transit of which is to be observed. Thus, 
suppose the index n to be at o, in the upper part of the circle, 
when the telescope is horizontal : then, by elevating the telescope, 
the index is moved downward. Suppose the position to be that 
represented in the figure, then the number of degrees between o 
and the index is the altitude. 

The wire plate placed in the focus of the transit telescope, has 
attached to it five vertical wires together with one horizontal wire. 
In order to be seen at night, these wires, or rather the field of view, 
requires to be illuminated by artificial light. The illumination of 
the field is effected by making one of the cones hollow, and ad- 
mitting the fight of a lamp placed in the pillar opposite the orifice ; 
which light is directed to the wires by a reflector placed diagonally 
in the telescope. The reflector, having a large hole in its centre, 



28 ASTRONOMICAL INSTRUMENTS. 

does not interfere with the rays passing down the telescope from 
the object.* 

The wires are seen as dark lines upon a bright ground. In 
some of the best instruments recently constructed there is a neat 
contrivance for illuminating the wires directly, so as to make them 
appear bright upon a dark ground, which is intended to be used in 
making observations upon faint stars. 

Sometimes the transit instrument is furnished with a meridian graduated circle 
of large size, designed to be used for the measurement of meridian altitudes or 
zenith distances. It then takes the name of Meridian Circle or Transit Circle; 
and serves for the determination of both the right ascension and declination of a 
heavenly body. The meridian circle of the observatory recently established at 
Pulkova, near St. Petersburg, has two meridian limbs, provided each with four 
reading microscopes. 

49. We will now explain the principal adjustments of the tran- 
sit. Upon setting the instrument up it should be so placed that 
the telescope, when turned down to the horizon, should point north 
and south, as near as can possibly be ascertained. This being 
done, then — 

(1.) To adjust the line of collimation. 

This adjustment consists in bringing the central vertical wire, 
within the telescope, to intersect the optical axis, which is sup- 
posed to be fixed by the maker of the instrument perpendicularly 
to the axis of rotation. There is no occasion with this instrument 
to have the horizontal wire intersect the optical axis with exact- 
ness. Direct the telescope to some small, distant, well-defined 
object, (the more distant the better,) and bisect it with the middle 
of the central vertical wire ; then lift the telescope out of its 
angular bearings, or Y's, and replace it with the axis reversed. 
Point the telescope again to the same object, and if it be still bi- 
sected, the collimation adjustment is correct ; if not, move the 
wires one half the angle of deviation, by turning the small screws 
that hold the wire plate, near the eye-end of the telescope, and the 
adjustment will be accomplished : but, as half the deviation may 
not be correctly estimated in moving the wires, it becomes neces- 
sary to verify the adjustment by moving the telescope the other 
half, which is done by turning the screw that gives the small azi- 
muth motion to the Y before spoken of, and consequently to the 
pivot of the axis which it carries. Having thus again bisected the 
object, reverse the axis as before, and if half the error was cor- 
rectly estimated, the object will be bisected upon the telescope 
being directed to it. If it should not be bisected, the operation of 
reversing and correcting half the error must be gone through again, 
and until after successive approximations the object is found to be 
bisected in both positions of the axis ; the adjustment will then be 
perfect.* 

* Woodhouse's Astronomy, vol. i. pp. 70-72 ; also Simm's Treatise on Mathe- 
matical Instruments, p. 53. 



TRANSIT INSTRUMENT. 29 

It is desirable that the central wire should be truly vertical, as 
we should then have the power of observing the transit of a star 
on any part of it, as well as the centre. It may be ascertained 
whether it is so, by elevating and depressing the telescope, when 
directed to a distant object : if the object is bisected by every part 
of the wire, the wire is vertical, (or rather it is perpendicular to 
the axis of rotation of the telescope, and becomes vertical so soon 
as the axis of rotation is made horizontal.) If it is not bisected, 
the wire should be adjusted, by turning the inner tube carrying 
the wire plate until the above test of its verticality be obtained. 

50. (2.) To set the axis of rotation of the telescope horizon- 
tal. This adjustment is effected by means of a spirit-level ; either 
attached to two upright arms bent at their upper extremities, by 
which it is hung on the two pivots of the axis, or else having two 
legs and standing upon the axis. In the first position it is called 
a hanging level, and in the second a riding level. At one end of 
the level is a vertical adjusting screw, by which that end may be 
elevated or depressed. Put the level in its place, and observe to 
which end of the level the bubble runs, which will always be the 
more elevated end ; bring it back to the middle by the Y screw for 
vertical motion, and take off the level and hang it on again with 
the ends reversed. Then, if the bubble is again found in the mid- 
dle, the level is already parallel to the axis, and the axis horizon- 
tal ; but if not, adjust one half the error by the adjusting screw of 
the level, and the other half by the Y screw ; and let the operation 
of reversing, and adjusting by halves, be repeated until the bubble 
will remain stationary in either position of the level, in which case 
both the level and axis will be horizontal. 

51 . (3.) To adjust the line of collimation to the plane of the me- 
vidian. We have said, that upon setting the instrument up, the 
telescope is to be brought into the meridian plane, as near as can 
be ascertained. One mode of establishing it, is to direct the tel- 
escope to the pole star, and by repeated observations find the 
position corresponding to its greatest or least altitude. At the 
present time, we may instead compute by means of existing tables 
founded on observation, the time of the meridian transit of the 
pole star, and at that computed time bisect the star by the middle 
vertical wire. Afterwards the line of collimation may be placed 
still more exactly in the plane of the meridian in the following 
manner : Note the times of two successive superior transits of the 
pole star across the central vertical wire, and the time of the inter- 
vening inferior transit. If the line of collimation were exactly in 
the plane of the meridian, as the diurnal circles are bisected by 
this plane, the interval between the superior and next inferior tran- 
sit would be precisely equal to the interval between the inferior 
and next superior transit. Accordingly, if these intervals are not 
in fact equal, find by repeated trials the position of the telescope 



30 ASTRONOMICAL INSTRUMENTS. 

and vertical wire for which triey are equal, and the line of collima 
tion will then be in the plane of the meridian. 

Instead of establishing this equality by a system of trial and 
error, w r e may, by means of a formula which has been investiga- 
ted for the purpose, compute from an observed inequality the 
amount of the movement in azimuth necessary to correct the error 
of position of the instrument. 

Another, and generally a more convenient method, is to observe one of the tran- 
sits of the pole star, and also the transit of some star that crosses the meridian near 
the zenith, and which follows or precedes the pole star by a known interval! (differ- 
ence of right ascensions of the two stars,) and compare the observed interval with 
the calculated interval. The difference of the two may be made to disappear by 
repeated trials : or a formula may easily be investigated, which shall make known 
the angular movement of the instrument necessary to make the observed and cal- 
culated intervals precisely equal. 

The method of regulating the clock required in making this ad- 
justment, will be explained when we come to treat of the astro- 
nomical clock. 

52. When the transit telescope has once been placed accurately 
in the meridian plane, in order to avoid the repetition of trouble- 
some verifications of its position, a meridian mark should be set 
up, and permanently established, at a distance from the instru- 
ment ; its place being determined by means of the middle or me- 
ridional wire. At Greenwich two such marks, one to the north 
and another to the south, are used ; they are vertical stripes of 
white paint upon a black ground, on buildings about two miles dis- 
tant from the observatory. The position of the telescope is verified 
by sighting at the meridian mark, when it is once established. 

53. The times of the transits of the heavenly bodies are ascer- 
tained as follows : in the case of a star, the moments of its cross- 
ing each of the five vertical wires are noted ; as the wires are 
equally distant from each other, the mean of these times (or their 
sum divided by 5) will be the time of the star's crossing the mid- 
dle wire, or of its meridian transit. The utility of having five 
wires, instead of the central one only, will be readily understood, 
from the consideration that a mean result of several observations 
is deserving of more confidence than a single one ; since the 
chances are that an error which may have been made at one wire 
will be compensated by an opposite error at another.* If the body 
observed has a disc of perceptible magnitude, as in the cases of the 
sun, moon, and planets, the times of the passage of both the west- 
ern and eastern limb across each of the five wires are noted, and 
the mean of the whole taken, which will be the instant of the me- 
ridian transit of the centre of the body. 

The time of the meridian transit of a body may, in this manner, 
be ascertained within a few tenths of a second. 

54. When a star is on the meridian, its declination circle (Def. 

* Simm's Mathematical Instruments, p. 59. 



ASTRONOMICAL CLOCK. 31 

16, p. 15) coincides with the meridian; moreover, the arc of the 
equator which lies between the declination circles of two stars, 
measures their difference of right ascension, (see def. 26, p. 17.) 
It follows, therefore, that in the interval between the transits of any 
two stars, the arc of the equator which expresses their difference 
of right ascension will pass across the meridian, the rate of the 
motion being that of 1 5° to a sidereal hour : hence the difference 
of the times of transit of two stars, as observed with a sidereal 
clock, when converted into degrees by allowing 15° to the hour, 
will be the difference between the right ascensions of the two stars. 
We may, then, in this manner, by means of a transit instrument 
and sidereal clock, find the differences between the right ascension 
of any one star and the right ascensions of all the others. This 
being done, as soon as the position of the vernal equinox with re- 
spect to the same star becomes known, (and we shall show how 
to find it,) the absolute right ascensions of all the stars will 
also become known. Thus RR', (Fig. 8,) is the difference of 
right ascension of the stars S and S', their absolute right as- 
censions being VR and VR', and VR is the distance of the vernal 
equinox V from the declination circle of the star S ; and it will at 
once be seen that if RR' be found, in the manner just explained 
so soon as VR becomes known, by adding it to RR' we shall have 
VR' the right ascension of the star S'. In the actually existing 
state of astronomical science, the right ascensions of all the stars 
are more or less accurately known, and a right ascension sought 
is now obtained directly, by noting the time of the transit of the 
body with a sidereal clock regulated so as to indicate Oh. Om. Os. 
when the vernal equinox is on the meridian, and converting it into 
degrees. 

ASTRONOMICAL CLOCK. 

55. The astronomical clock is very similar to the common clock. 
It has a compensation pendulum ; that is, a pendulum so construct- 
ed that its length is unaffected by changes of temperature. The 
hours on the face are marked from 1 to 24. 

56. Astronomers make use of sidereal time (as already stated) 
in determining the right ascensions of the heavenly bodies, but for 
all other purposes they generally use mean solar time. 

57. To regulate a sidereal clock. — When a clock is used for de- 
termining differences of right ascension, (54,) it is adjusted to side- 
real time if it goes equally and marks out 24 hours in a sidereal 
day ; it being altogether immaterial at what time it indicates Oh. 
Om. Os. To ascertain the daily rate of going of a clock which is 
to be adjusted to sidereal time for the purpose just mentioned, note 
by the clock the times of two successive meridian transits of the 
same star : the difference between the interval of the transits and 
24 hours will be the daily gain or loss (as the case may be) of the 



33 ASTRONOMICAL INSTRUMENTS. 

clock with respect to a perfectly accurate sidereal clock.* If the 
gain or loss, when found after this manner, proves to be the same 
each day, then the mean rate of going is the 'same each day. 

Next, to be able to discover the rate from hour to hour during the day, it is ne- 
cessary to have obtained beforehand, at various times, and under various states of 
the circumstances likely to influence the rate of going of the clock, the differences 
between the times of the transits of a number of different stars, (correcting propor- 
tionally for the daily rate,) and to take the mean of the several differences found for 
each pair of stars for the exact difference of their transits. When this lias been 
done, the rate of the clock may be found at all hours during the day by noting by 
the clock the differences between the times of the transits of these stars, and com. 
paring these with the exact differences already found. At the present time, the 
right ascensions of the stars being known, to ascertain the rate from hour to hour, 
we have only to compare the intervals of time given by the clock between the 
transits of different stars taken in the order of their right ascension with their 
differences of right ascension. 

58. The sidereal clocks now in use are made to indicate Oh. 
Om. Os. when the vernal equinox is on the superior meridian. For 
the regulation of such clocks, it is necessary to know not only their 
rate but also their error. This is found by noting the time of the 
transit of a star, and comparing this with its right ascension ex- 
pressed in time. If the two are equal the clock is right, otherwise 
their difference will be its error. 

If the error of the rate of a clock be considerable, it should be 
diminished by altering the length of the pendulum ; otherwise, it 
may be allowed for. The stars best adapted to the regulation of 
clocks are those in the vicinity of the equator ; for, as their motion 
is more rapid than that of the stars more distant from the equator, 
there is less liability to error in noting their transits. 

59. A mean solar clock is usually regulated by observations up- 
on the sun. The method of regulating it cannot be adequately 
explained until we have treated of the apparent motion of the sun. 
It will here suffice to state, that with the instruments we have now 
described, the sun's motion can be ascertained ; and therefore, as 
a knowledge of this is all that is necessary in order that we may 
be able to obtain the mean solar time at any instant, that it is pos- 
sible to express all intervals of time in mean solar time. 

ASTRONOMICAL CIRCLE. 

60. An Astronomical Circle is an instrument designed for the 
measurement of the zenith distances or altitudes of the heavenly 
bodies at the instants of their arrival on the meridian. Its essen- 
tial parts are a graduated circular limb, a telescope turning upon a 
horizontal axis which passes through the centre of the limb, and a 
micrometer microscope, (40,) or other piece of apparatus, for read- 
ing off the angles upon the limb. It is sometimes mounted 
upon an upright stem, which either turns upon fixed supports or 

* It is not necessary, in order to obtain the daily rate of a sidereal clock, that 
the transit instrument should be adjusted to the plane of the meridian. It is only 
requisite that it should be kept fixed in some one vertical plane. 



MURAL CIRCLE. 



33 



rests upon a tripod, and can be turned around in azimuth ; but, in 
general, the larger circles in the best furnished observatories have 
their axis let into a massive pier, or wall, of stone, and capable 
of only such small motions in the horizontal and vertical directions, 
under the action of screws, as maybe necessary for its adjustment 
to the horizontal position and perpendicular to the meridian plane. 
These are called Mural Circles. For greater accuracy the angle 
is read off at six different points of the limb by means of six sta- 
tionary micrometer microscopes, and the mean of the different 
readings taken for the angle required. Fig. 16 is a side view of a 
mural circle. The graduation is on the outer rim, which is per- 
pendicular to the plane of the wall CDFE. One of the reading 

Fig. 16. 




microscopes is represented at A. The others, which are omitted 
in the figure, are disposed at equal distances around the rim. The 
position of the telescope may be changed by unclamping it and 
clamping it to a different part of the limb. In taking an angle, the 
telescope is made fast to the limb, and the limb glides past the sta- 
tionary microscopes. 

The six reading microscopes, together with the power of chang- 
ing the position of the telescope on the limb, so as to read off the 
angle from all parts of the limb, when the mean results of a great 
number of observations are taken, do away with, or at least very 
considerably lessen the errors of graduation, centring, and une- 
qual expansion. 

5 



34 ASTRONOMICAL INSTRUMENTS. 

61. In place of mural circles, Mural Quadrants have been much used. Sine© 
the mural quadrant has its graduated limb only one fourth the size of the limb of 
the mural circle, it can be made larger than the circle. But the circle is better 
balanced than the quadrant, and the quadrant does not possess the advantages 
which have been enumerated as resulting in the case of the circle from the use of 
a number of reading microscopes, and from the power to change the position of 
the telescope on the limb. Besides, two mural quadrants, one to observe the stars 
north of the zenith, and another to observe the stars south of the zenith, are neces. 
sary to effect the general object, accomplished by one mural circle, of ascertaining 
the zenith distance or altitude of any heavenly body at the time of its arrival on 
the meridian. 

62. The largest astronomical circles that have yet been con- 
structed, are to be found, it is said, in the Dublin and Cambridge 
Observatories. That in the Dublin Observatory is 8 feet in diame- 
ter, and has an azimuth motion, (that is, a motion about a vertical 
axis.) The other is a mural circle. The mural circle in the Na- 
tional Observatory at Washington has a diameter of a little more 
than 5 feet. 

The large mural quadrants of the Greenwich Observatory are of 8 feet radius. 

63. There is another modification of the astronomical circle, called the Zenith 
Sector, which is used to measure the meridian zenith distances of stars that cross 
the meridian within a few degrees of the zenith. The limb extends only about 
10 Q on each side of the lowest point. It can, accordingly, be made larger than the 
limb of the circle or quadrant. The zenith sector in the observatory at Greenwich 
has a radius of 12 feet. 

64. The mural circle, like the transit instrument, requires three 
adjustments : 1 . Its axis must be made horizontal ; 2. Its line of 
collimation (38) must be made perpendicular to the horizontal axis; 
3. The line of collimation must be made to move in the plane of 
the meridian. 

A simple mechanical contrivance exists for carrying the first of 
the adjustments into complete effect. When the axis is made ho- 
rizontal, the line of collimation describes a vertical circle ; but it 
may describe a small circle of the celestial sphere. To make it ne- 
cessarily describe a great circle, and a meridional circle, there are 
no mechanical means. Astronomical ones must be resorted to ; 
and even with those, the two latter adjustments are not accom- 
plished without great difficulty. We may, on this occasion, use 
the transit. When a star is on the meridional wire of the 
transit instrument, so move the mural circle that the star may 
be on its middle wire. Next, observe by the transit instrument 
when a star, on, or very near to the zenith, crosses the meridian : 
if, at that time, the star is on the middle vertical wire of the tele- 
scope of the mural circle, then its line of collimation is rightly 
adjusted. If the star is on the middle wires of the two telescopes 
at different times, note their difference and adjust accordingly.* 

65. The horizontal point of the limb, technically so called, is 
the place of the index (or centre of the microscope) answering to 

* This adjustment must be conducted by some formula which expresses the re- 
lation between the difference of the times, and the inclination of the line of collima- 
tion to the plane of the meridian, (Woodhouse's Astronomy, p. 117.) 



MURAL CIRCLE. 35 

a horizontal position of the line of collimation of the telescope. 
Perhaps the simplest method of obtaining this point is the follow- 
ing : Direct the telescope upon some star at the moment of its 
culmination, and read oft the angle on the limb. Procure an arti- 
ficial horizon, (see art. 79,) and on the following night direct the 
telescope upon the image of the same star, as seen in the artificial 
horizon. By the laws of reflexion, the angle of depression of this 
image will be equal to the angle of elevation of the star. Accord- 
ingly the arc on the limb which passes before the reading micro- 
scope, in moving the telescope from the star to its image, will be 
double the altitude of the star, and its point of bisection the hori- 
zontal point.* This point may also be found by directing the 
telescope upon a star whose altitude is known. 

66. In the case of the mural quadrant, if there is no altitude that can be relied 
on as having been obtained with ail attainable accuracy, it is necessary to have 
recourse to the zenith sector. This instrument is so constructed and arranged, 
that its horizontal axis can be reversed in position. By taking the zenith distance 
of a star with its face towards the east, and then of the same star with the face to- 
wards the west, the half sum of the two will be its true zenith distance. With this 
we may readily find the vertical point, and thence the horizontal point, on the limb 
of the mural quadrant, by directing the telescope upon the star observed with the 
sector, when it is on the meridian. 

67. The adjustments of the mural circle having all been effect- 
ed, and the horizontal point determined, if the instrument be set t<? 
this point, and the telescope afterwards directed upon any star in 
the meridian, the arc of the limb that passes by the reading mi- 
croscope will be the altitude of the star. In making the observa- 
tion the telescope must be brought into such a position that the 
star will be bisected by the horizontal wire, as it passes through 
the field of view. The altitude of the sun, moon, or any planet, 
may be ascertained by measuring the altitudes of the upper and 
lower limbs, and taking their half sum for the altitude of the cen- 
tre : or, if the apparent semidiameter be known, by adding this to 
the altitude of the lower limb, or subtracting it from the altitude of 
the upper limb. 

68. The meridian altitude or zenith distance of a heavenly body 
having been measured with an astronomical circle, or other similar 
instrument, at a place the latitude of which is known, its declina- 
tion may easily be found. For, let s, (Fig. 10,) represent the point 
of meridian passage of a star, or other heavenly body, which crosses 
the meridian to the north of the zenith (Z.) Es will be its declina- 
tion, (Def. 27, p. 17,) Zs its meridian zenith distance, and ZE the 
latitude of the place of observation (0,) (Def. 33, p. 18 :) and we 
obviously have 

Es = ZE + Zs . . . (a). 

* The method of using the level for the determination of the horizontal point 
may be found explained in Herschel's Astronomy, p. 93. Another piece of appa- 
ratus, used for the same purpose, called the Floating Collimator, is described in the 
same work, p. 95. 



36 ASTRONOMICAL INSTRUMENTS 

If the star cross the meridian at some point s' between the ze~ 
nith (Z) and the equator (E,) we shall have E$' = ZE-— Zs', (6) - r 
and if its point c£ transit be some point s" to the south of the equa- 
tor (E,) we shall have Es" = Zs" — ZE, and — Es" = ZE — 
Zs", (c). The three formulae («), (6), and (c), may all be compre- 
hended in one, viz : 

Declination = latitude + meridian zenith distance , , , (1) 
if we adopt the following conventional rules : 1 , North latitude is 
always positive \ 2. The zenith distance is positive when it is 
North, that is r when the star is north of the zenith, and negative 
when it is South ; 3. The declination is North if it comes out posi- 
tive, and South if it comes out negative. 

If the latitude is South, it must be regarded as negative r and the zenith distance 
must be affected with the minus sign when it is South, and with the plus sign when 
it is North. The rule for the declination is the same. In general, North latitude is 
-f-, South latitude — . The zenith distance has the same sign as the latitude when 
it is of the same name, the contrary sign when it is of a contrary name. North 
declination is -J-» South declination — . 

The latitude which is here supposed to be known, may be found 
by measuring (67) the meridian altitudes of a circumpolar star at 
its inferior and superior transits, and taking their half sum. For, 
as the pole lies midway between the points at which the transits 
take place, its altitude will be the arithmetical mean, or the half 
sum of the altitudes of these points, and the altitude of the pole is 
equal to the latitude of the place, (34.) 

69. When the right ascension and declination of a heavenly body 
have been obtained from observation, with a transit instrument and 
circle, (54, 68,) its longitude and latitude may be computed. For, 
let S (Fig, 8) represent the place of the body, VRQE the equa- 
tor, VLTW the ecliptic, and P, K, the north poles of the equator 
and ecliptic. In the spherical triangle PKS we shall know PS 
the complement of SR the declination, and the angle KPS = 
ER = EV -f- VR = 90° + right ascension ; and if we suppose the 
obliquity of the ecliptic to be known, we shall know PK, We 
may therefore compute KS, and the angle PKS. But KS is the 
complement of SL, which is the latitude of the body S ; and 
PKS = 180° — EKS = 180° — (WV + VL) = 180° — (90° + 
longitude) = 90° — longitude. 

The obliquity of the ecliptic, which we have here supposed to 
be known, is, in practice, easily found ; for it is equal to TQ, the 
sun's greatest declination. 

ALTITUDE AND AZIMUTH INSTRUMENT. 

70. The Altitude and Azimuth Instrument consists, essentially, 
of a telescope with two graduated limbs, the one horizontal and tne 
other vertical. The telescope turns about the centre of the verti- 
cal limb, or turns with the limb about its centre ; and the ver- 
tical limb turns, with the telescope, about the vertical axis of the 
horizontal limb. 



EQUATORIAL 



37 




If the telescope be brought into the meridian plane, and after- 
wards directed upon a star out of this plane, the arc of the hori- 
zontal limb passed over by the index will be the azimuth of the 
star. The vertical limb will serve to measure its altitude* 

71. The Meridian Line (Def. 8, p 14) at a place may easily be 
determined with the altitude and azimuth instrument, by a method 
called the Method of Equal Alti- Fig. u, 

tudes. Let (Fig. 17) represent 
the place of observation, NPZ the 
meridian, and S, S' two positions 
of the same star, at which the alti- 
tude is the same. Now, the spher- 
ical triangles ZPS and ZPS' have 
the side ZP common, ZS=ZS', ^ 
and (allowing the stars to move in 
circles) P S =P S'. Hence they are 
equal, and consequently the angle 
PZS=PZS'; that is, equal altitudes of a star correspond to equal 
azimuths. Therefore, by bisecting the arc of the horizontal limb, 
comprehended between two positions of the vertical limb for which 
the observed altitude of a star is the same, we shall obtain the me- 
ridian line. 

The meridian line may be approximately determined by this method with the 
common theodolite ; the observations being made upon the sun. The result will 
be more accurate if they be made towards the summer or winter solstice, when the 
sun will have but a slight motion towards the north or south in the interval of the 
observations. It is, however, easy to determine and allow for the effect -of the sun's 
•change of place in the heavens. 

72. When the time is accurately known, the north and south line may be found 
very easily by directing the telescope of any instrument that has a motion in azi- 
muth upon a star in the vieinity of the pole and at a distance from the zenith, at 
the moment of its arrival on the meridian, (which, as will be understood in the se- 
quel, can now easily be determined from existing data.) 

EQUATORIAL. 

73. The Equatorial is similar, in its construction, to the altitude 
and azimuth instrument It is so called from the circumstance of 
one of the limbs being placed in a position parallel to the plane of 
the equator. The axis of this limb is then parallel to the axis of 
the heavens ; and the other limb, to the centre of which the tele- 
scope is attached, is parallel in every one of its positions to the 
plane of some one celestial meridian. The limb which is parallel 
to the equator serves for the measurement of differences of right 
ascension, and the other for the measurement of declinations. The 
equatorial is regarded as one of the most indispensable instruments 
of an astronomical observatory. It is particularly useful in the 
measurement of apparent diameters, and in all observations that 
require the telescope to be directed upon a body for a considerable 
period of time ; as, by giving the limb to which the telescope is 
attached a slow motion from east to west, the body may be follow 



3S 



ASTRONOMICAL INSTRUMENTS, 



ed in its diurnal motion, and kept continually within the field of 
view. This motion is generally produced by clock-work, without 
the use of the hand. 

It is also frequently used for determining the right ascension and declination of a 
comet, or other heavenly body, which for some reason cannot, at the time, be ob- 
served in the meridian ; and for finding and obtaining a protracted view, or fixing 
more accurately the place of an object invisible to the'naked eye, whose place has 
been approximately calculated from the results of previous observations. Another 
important object to which it may be applied, is the determination of small differ- 
ences of right ascension and declination, and thus of the relative positions of con- 
tiguous objects. Its determinations of declinations and differences of right ascen- 
sion, in general, are to be deemed less accurate than those effected with the mural 
circle and transit instrument ; as, from its more complicated structure, and peculiar 
position, the parts have less stability and are more subject to unequal strains, 
bendings, and expansions, than those of the instruments just named. 

74. The adjustments of the equatorial are somewhat complica- 
ted and difficult. They are best performed by following the pole- 
star round its entire diurnal circle, and by observing, at proper 
intervals, other considerable stars whose places are well ascertained. 
(Herschel.) 

75. In addition to the instruments that have now been de- 
scribed, which are designed and used for the measurement 
ef the angular distances of bodies from some fixed point or 
circle in the heavens, astronomers have found it convenient 
and important to have another instrument, or piece of appa- 
ratus, with which to determine directly the relative situation 
of two stars that are near to each other ; so near as to be 
seen, at the same time, in the same field of view. The ap- 
paratus used for this purpose is attached to the telescope of 
the equatorial, or other instrument, and is called a Micrometer. 
Another important use to which it is put, is the measurement 
of the apparent diameters of the heavenly bodies. It has a 
variety of forms. The simplest is known by the name of the 
Wire-Micrometer. It is placed in the focus of the telescope. 
It consists of two forks of brass^ bb'b, cc'c, (Fig. 18) sliding 
one within the other, and having eaeh a very fine wire, or 
spider-line, e, and d, stretched perpendicularly across from 
one prong to the other. These forks are placed length- 
wise in a shallow rectangular box, aa'aa', about 2 inches 
wide and 4 inches long ; and have each fine-threaded mi- 
crometer screws, f, f, working against the ends, b', and c'. 
The graduated heads of these screws are not represented in 
the figure, but they may be seen in Fig. 19. They pass 



Fig 


.18. 


'i\-^-j 


mm? 


b'- 


1 , 




Id 




c 


S c 


a * 




I 

I 


I 


c 




1. 




'i- 


a'rzn 



Fig. 19. 



through the ends a', a', of the box, 
and have their graduated heads on 
the outside of it. Between the ends 
b', c' of the two forks and the con- 
tiguous ends a 1 , a' of the box are two 
spiral springs, h, h, which keep the 
ends of the forks firmly pressed 
against the ends of the screws, and 
draw the forks outward and the 
wires further apart whenever the 
screws are loosened. By turning 
the screws in the opposite direction 
the forks are pushed forward, and 
the wires brought nearer to each 
other. The number of complete turna and parts of a turn made by each screw,, as. 




SEXTANT. 39 

shown by its graduated head, will make known the fraction of an inch through 
which the end of it and the contiguous wire is moved. The screws can be so del. 
icately cut that they will measure with accuracy the yo-g^ of an inch. The linear 
space thus measured in the focus of the telescope must be converted into the equiv- 
alent angular space in the heavens. This is effected by fixing upon two contigu- 
ous stars, whose distance is accurately known, and measuring with the micrometer 
the linear distance of their images formed in the focus. In this way will be found 
how many seconds of angular space correspond to a given movement of either of 
the wires, as measured by the micrometer scale. The micrometer box is fastened 
perpendicularly aeross the eye-end of the tube of the telescope. The eye-piece 
of the telescope screws into the outer face of the box, (see Fig. 19,) and on looking 
into it, the wires d, e within the box are seen in its focus ; where also the images 
of the stars, formed by the object-glass, fall. To save the necessity of counting 
the revolutions of the micrometer screws, a linear scale is placed within the box, 
and at one side of it, consisting of a series of teeth, with intervening notches. This 
is represented in the diagram, (Fig. 18.) A motion of the wire from one notch to 
another answers, say, to one turn of the screw, and to 1' in space. 

To measure the angular distance of two stars, the wires are both brought into 
coincidence at the zero of this scale, when we will suppose that they fall between 
the stars. By turning the screws they are moved from this position, and the mo- 
tion is continued until the one star is accurately bisected by one wire, and the other 
star by the other wire. The number of notches which the wires have passed will 
express the number of minutes in the space between the stars ; to these are to be 
added the seconds answering to the fractional parts of a revolution, as shown by 
the divided heads of the screws. It will be seen, that in order to obtain the real 
distance between the two stars, the two wires d and e must be brought into such 
a position as to be perpendicular to the line of the stars. This is effected by giving 
to the whole box a revolving motion about the optical axis of the telescope, and 
bringing the wire I, which is perpendicular to d and e, into such a position as to 
bisect both the stars. The diameter of a heavenly body is measured in a similar 
manner ; the wires being brought into contact with the opposite limbs. 

76. To measure the angle made by the line of direction of two stars with a fixed 
line passing through one of them, it is necessary that the micrometer box should 
not only have a revolving motion around the axis of the telescope, but also a grad- 
uated circle to measure its amount. The cross-wire I is brought by this motion 
into coincidence — first with one line and then with the other, and then the an- 
gle read off. In this way may be found the angle made by the line of direction of 
two contiguous stars with the meridian, or a line perpendicular to the meridian, 
at the moment one of them is crossing this circle. This angle is called the Angle 
of Position of the two stars, and the micrometer that serves to measure it is called 
a Position Micrometer. The position of the wire I when perpendicular to the meri- 
dian may be found by turning it until one of the stars runs along the wire, while 
the telescope of the equatorial is stationary. Fig. 19 represents a position microm- 
eter. The micrometer box b, with its attached eye-piece c, is connected with the 
circle a, and is turned around with it by the small milled-head screw s, which 
works on an interior toothed wheel, and the angle is read off upon the stationary 
graduated circle above a, by aid of the vernier, moveable with the plate a. 

SEXTANT. 

77. The instruments which have now been described are ob- 
servatory instruments, the chief design of whose construction is to 
furnish the places of the heavenly bodies with all attainable exact- 
ness. That of which we are now to treat is much less exact, 
though still of great utility in determining the essential data of 
some of the practical applications of astronomical science ; as 
finding the latitude and longitude of a place, and the time of day: 
and is used chiefly by navigators, and astronomical observers on 
land, who are precluded, by their situation or other circumstances, 



40 



ASTRONOMICAL INSTRUMENTS. 



from using the more accurate instruments of an observatory. It is 
much more conveniently portable than any of these, and has not to 
be set up and adjusted at every new place of observation. Besides, 
as it is held in the hand, it can be used at sea, where, by reason 
of the agitations of the vessel, no instrument supported in the or- 
dinary way is of any service. 

78. The sextant may be defined, in general terms, to be an in- 
strument which serves for the direct admeasurement of the angu- 
lar distance between any two visible points. The particular quan- 
tities that may be measured with it, are, 1st, the altitude of i 
heavenly body ; 2d, the angular distance between any two visible 
objects in the heavens/ or on the earth. Its essential parts are a 
graduated limb BC, (Fig. 20,) comprising about 60 degrees of the 
entire circle, which is attached to a triangular frame BAC ; two 
mirrors, of which one (A) called the Index Glass, is moveable in 
connection with an index G about A the centre of the limb, and 
Fig. 20. the other (D) called the Ho- 

rizon Glass, is permanently 
fixed parallel to the radius 
AC drawn to the zero point 
of the limb, and is only half- 
silvered, (the upper half be- 
ing transparent ;) and an im- 
moveable telescope at E, 
directed towards the horizon- 
glass. The principle of the 
construction and use of the 
sextant may be understood 
from what follows : A ray of 
light SA from a celestial ob- 
jects, which impinges against 
the index-glass, is reflected 
off at an equal angle, and 
striking the horizon-glass (D) 
is again reflected to E, where the eye likewise receives through 
the transparent part of that glass a direct ray from another point 
or object S'. Now, if AS' be drawn, directed to the object S', 
SAS', the angular distance between the two objects S and S', is 
equal to double the angle CAG measured upon the limb of the 
instrument, (AC being parallel to the horizon-glass.) For, when 
the index-glass is parallel to the horizon-glass, and the angle on the 
limb is zero, AD, the course of the first reflected ray, will make 
equal angles with the two glasses, and therefore the angle SAD 
will become the angle S'AD, (= ADE ;) and the observer, look- 
ing through the telescope, will see the same object S' both by 
direct and reflected light. Now, if the index-glass be moved from 
this position through any angle CAG, the angle made by the re- 
flected ray which follows the direction AD with this glass, will be 




SEXTANT. 41 

diminished by an amount equal to this angle ; for, we have DAG = 
DAC — CA(j. Therefore the angle made with the index-glass 
by the new incident ray SA, which after reflexion now pursues the 
same course ADE, and reaches the eye at E, as it is always equal 
to that made by the reflected ray, will be diminished by this amount. 
Consequently, the incident ray in question will, on the whole, that 
is, by the diminution of its inclination to the mirror by the angle 
C AG and by the motion of the mirror through the same angle, be 
displaced towards the right, or upward, an angle S'AS equal to 
2GAC. Thus, the angular distance SAS' of two objects S, S', 
seen in contact, the one (S') directly, and the other (S) by reflex- 
ion from the two mirrors, is equal to twice the angle CAG that 
the index-glass is moved from the position (AC) of parallelism to 
the horizon-glass. 

Hence the limb is divided into 120 equal parts, which are called 
degrees ; and to obtain the angular distance between two points, 
it is only necessary to sight directly at one of them, and then 
move the index until the reflected image of the other is brought 
into contact with it ; the angle read off on the limb will be the an- 
gle sought. 

To obtain the angular distance between two bodies which 
have a sensible diameter, bring the nearest limbs into contact, and 
to the angle read off on the limb add the sum of the apparent semi- 
diameters of the two bodies, or bring the farthest limbs into con- 
tact, and subtract this sum. 

79. The sextant is also employed to take the altitude of a heav- 
enly body. A horizontal reflector, called an Artificial Horizon, 
is placed in front of the observer : the angle between the body and 
its reflected image is then measured, as if this image were a real 
object ; the half of which will be the altitude of the body. 

A shallow vessel of mercury forms a very good artificial horizon. 

In obtaining the altitude of a body, at sea, its altitude above 
the visible horizon is measured, by bringing the lower limb into 
contact with the horizon. To this angle is added the apparent 
semi-diameter of the body, and from the result is subtracted the 
depression of the visible horizon below the horizontal line, called 
the Dip of the Horizon. 

80. Hadley's Quadrant differs from the sextant in having a 
graduated limb of 45°, instead of 60°, in real extent, and a sight 
vane instead of a small telescope. It is not capable, then, of meas- 
uring any angle greater than about 90°, while the sextant will 
measure an angle as great as 120°, or even 140°, (for the gradua- 
tion generally extends to 140°.) The quadrant is also inferior to 
the sextant in respect to materials and workmanship, and its meas- 
urements are less accurate. 

6 



42 ASTRONOMICAL INSTRUMENTS. 



ERRORS OF INSTRUMENTAL ADMEASUREMENT. 

81. Whatever precautions may be taken, the results of instru- 
mental admeasurement will never be wholly free from errors. 
Errors that arise from inaccuracy in the workmanship or adjust- 
ment of the instrument may be detected and allowed for. But 
errors of observation are obviously undiscoverable. Since, how- 
ever, the chances are that an error committed at one observation 
will be compensated by an opposite error at another, it is to be 
expected that a more accurate result will be obtained if a great 
number of observations, under varied circumstances, be made, 
instead of one, and the mean of the whole taken for the element 
sought. And accordingly, it is the uniform practice of astronomi- 
cal observers to multiply observations as much as is practicable. 

TELESCOPE. 

82. An observatory is not completely furnished unless it is supplied with a large 
telescope for examining the various classes of objects in the heavens, and one or 
more smaller ones for exploring the heavens and searching for particular objects 
invisible to the naked eye, as faint comets, and making observations upon occa- 
sional celestial phenomena, as eclipses of the sun and moon, occultations of the 
stars, &c. Telescopes are divided into the two classes of Reflecting and Refract- 
ing Telescopes. In the former class the image of the object is formed by a con- 
cave speculum, and in the latter by a converging achromatic lens. This image is 
viewed and magnified by an eye-glass, or rather by an achromatic eye-piece 
consisting of two glasses. In the simplest form of the reflecting telescope, the 
Herschelian, the image formed by the concave speculum is thrown a little to one 
side, and near the open mouth of the tube, where the observer views it through the 
eye-glass, with his back turned towards the object. 

83. The magnifying power of a telescope is to be carefully distinguished from its 
illuminating and space-penetrating power. A telescope magnifies by increasing 
the angle under which the object is viewed : it increases the light received from 
objects, and reveals to the sight faint stars, nebulae, &c, by intercepting and con- 
verging to a point a much larger beam of rays. The magnifying power is meas- 
ured by the ratio of the focal length of the object-glass, or speculum, to that of the 
eye-piece. The illuminating and space-penetrating power, (for faint objects,) if 
we leave out of view the amount of light lost by reflexion and absorption, is meas- 
ured by the proportion which the aperture of the object-glass or speculum bears to 
the pupil of the eye. Telescopes are provided with several eye-glasses of various 
powers. The power to be used varies with the object to be viewed, and the purity 
and degree of tranquillity of the atmosphere. Of two telescopes of the same focal 
length, that which has the largest aperture will form the brightest image in the 
focus, and therefore, other things being equal, admit of the use of the most power- 
ful eye-piece. In this way it happens that the available magnifying power indi- 
rectly depends materially upon the size of the aperture. In all telescopes there is 
a certain fixed ratio between the aperture and focal length, or at least limit to this 
ratio. In reflecting telescopes it is about one inch of aperture for every foot of focal 
length, and in refracting, one inch of aperture for from one to two feet of focal 
length. Reflectors and refractors of the same focal length have about the same 
actual magnifying and illuminating power. The highest available magnifying 
power that has yet been obtained is about 6,000 ; but this" was applicable only to 
the faintest stars and nebulous spots. With the best telescopes a magnifying power 
of a few hundred is the highest that can be applied to the moon and planets. The 
largest reflecting telescope that has yet been constructed, and directed to the heav- 
ens, is the celebrated one of Sir William Herschel, of 40 feet focus, and 4 feet 
aperture. Its illuminating power was about 35,000, which makes its space-pene 



CORRECTIONS OF CO-ORDINATES. 43 

trating power nearly 190 times the distance of the faintest star visible to tho naked 
eye ; and its highest magnifying power was 6,450 * The most powerful refractor 
yet constiucted is in the new observatory at Pulkova, near St. Petersburg. It has 
an aperture of very nearly 15 inches, (14.93 inches,) and a focal length of 22 feet. 
The best telescope in the United States is the refractor in the new observatory at 
Cincinnati.f Its aperture is 12 inches, and focal length about 17 feet. The 
field of view of telescopes diminishes in proportion as the magnifying power in- 
creases. It s stated that with a magnifying power of between 100 and 200 it is 
a circle not as large as the full moon ; and with a power of 600 or 1000 is nearly 
filled by one of the planets, while a star will pass across it in from two to three 
seconds. 

84. The diminution of the field of view, and the trepidations of the image oc- 
casioned by the varying density of the atmosphere, and the unavoidable tremors of 
the instrument, must ever affix a practical limit to the magnifying power of tele- 
scopes. This limit, it is probable, is already nearly attained, for the highest pow- 
ers of the best telescopes can now be used only in the most favorable states of the 
weather.:}: 

85. The large refracting telescopes are equatorially mounted, that they may, as 
readily as possible, be directed and retained upon an object. 

86. "The small telescope, called a comet-seeker ; is a refractor of large aperturo 
and wide field. Its power does not exceed 100. 



CHAPTER HI. 



ON THE CORRECTIONS OF THE CO-ORDINATES OF THE OBSERVED 
PLACE OF A HEAVENLY BODY. 

87. Angles measured at the earth's surface with astronomical 
instruments, answer to the Apparent Place of a heavenly body, 
and are termed Apparent elements. In astronomical language, 
the True Place of a heavenly body is its real place in the heavens, 
as it would be seen from the centre of the earth. Angles which 
relate to the true place are denominated True elements. The 
apparent co-ordinates of a star are reduced to the true, by the ap- 
plication of certain corrections, called Refraction, Parallax, and 
Aberration. 

88. Refraction and aberration are corrections for errors com- 

* A reflecting telescope, inferior to Herschel's in size, the diameter of the spe- 
culum being 3 feet, and the focal length 26 feet, but pronounced by Dr. Robinson 
superior to it in defining power, has, within a few years, been constructed by the 
Earl of Rosse, of Ireland. The same nobleman has just completed the construc- 
tion of a reflecting telescope of unparalleled dimensions, from the use of which im- 
portant discoveries may be anticipated. The diameter of the speculum is 6 feet, 
and it has a focal length of 53 feet. 

\ See Note II. _ 

% The illuminating and space-penetrating power of telescopes may, however, yet 
be greatly increased, and a greater distinctness and definiteness in the outline of 
objects may be obtained. Much may perhaps be gained also by setting up an ob- 
servatory on the top or sides of some lofty mountain above the greater impurities 
and disturbances of the lower regions of the atmosphere, and under a tropical sky. 



44 CORRECTIONS OF CO-ORDINATES. 

« 

mitted in the estimation of a star's place, while parallax serves to 
transfer the co-ordinates from the earth's surface to its centre. The 
object of the reduction of observations from the surface to the cen- 
tre of the earth, is to render observations made at different places 
on the earth's surface directly comparable with each other. Ob- 
servers occupying different stations upon the earth refer the same 
body (unless it be a fixed star) to different points of the celestial 
sphere. Their observations cannot, therefore, be compared to- 
gether, unless they be reduced to the same point, and the centre 
of the earth is the most convenient point of reference that can be 
chosen. 

89. The co-ordinate planes or circles to which the place of a 
star is referred, (p. 17,) are not strictly stationary, but, on the con- 
trary, have a continual slow motion with respect to the stars. 
Hence, the true co-ordinates of a star's place which have been 
found for any one epoch, will not answer, without correction, for 
any other epoch. The reduction from one epoch to another is 
effected by applying two corrections, called Precession and Nu- 
tation. 

REFRACTION. 

90. We learn from the principles of Pneumatics, as well as by 
experiments with the barometer, that the atmosphere gradually 
decreases in density from the earth's surface upward. We learn 
also from the same sources, that it may be conceived to be made 
up of an infinite number of strata of decreasing density, concentric 
with the earth's surface. From the known pressure and density 
of the atmosphere at the surface of the earth, it is computed, that 
by the laws of the equilibrium of fluids, if its density were through- 
out the same as immediately in contact with the earth, its altitude 
would be about 5 miles. Certain facts, hereafter to be mentioned, 
show that its actual altitude is not far from 50 miles. Now, it is 
an established principle of Optics, that light in passing from a va- 
cuum into a transparent medium, or from a rarer into a denser 
medium, is bent, or refracted, towards the perpendicular to the 
surface at the point of incidence. It follows, therefore, that the 
light which comes from a star, in passing into the earth's atmo- 
sphere, or in passing from one stratum of atmosphere into another, 
is refracted towards the radius drawn from the centre of the earth 
to the point of incidence. 

91. Let MmnN, NrcoO, OoqQ, (Fig. 21,) represent successive 
strata of the atmosphere. Any ray Sp will then, instead of pur- 
suing a straight course Spx, follow the broken line pabc ; being 
bent downward at the points p, a, b, c, &c, where it enters the 
different strata. But, since the number of strata is infinite, and 
the density increases by infinitely small degrees, the deflections 
apx, bay, &c, as well as the lengths of the lines pa, ab, &c, are 



REFRACTION. 



45 



infinitely small; and Fig. 21. 

therefore pabc, the 
path of the ray, is a 
broken line of an in- 
finite number of parts, 
or a curved line con- 
cave towards the 
earth's surface, as it 
is represented in Fig. 
22. Moreover, it lies 
in the vertical plane 
containing the origi- 
nal direction of the 
ray ; for, this plane is 
perpendicular to all 
the strata of the atmosphere, and therefore the ray will continue 
in it in passing from one to the other. 

92. The line OS 7 (Fig. 22) drawn tangent to paO, the curvilinear 
path of the light, at its lowest point, will represent the direction in 
which the light enters the eye, and therefore the apparent line of 

Fig. 22. 





direction of the star. If, then, OS be the true direction of the star, 
the angle SOS' will be the displacement of the star produced by 
Atmospherical Refraction. This angle is called the Astronomical 
Refraction, or simply the Refraction. 

Since paO is concave towards the earth, OS' will lie above OS ; 
consequently, refraction makes the apparent altitude of a star 
greater than its true altitude, and the apparent zenith distance of 
a star less than its true zenith distance. (We here speak of the 
true altitude and true zenith distance, as estimated from the station 
of the observer upon the earth's surface.) Thus, to obtain the 
true altitude from the apparent, we must subtract the refraction ; 
and to obtain the true zenith distance from the apparent, we must 
add the refraction. As refraction takes effect wholly in a vertical 
plane, (91,) it does not alter the azimuth of a star. 



46 



CORRECTIONS OF CO-ORDINATES. 



93. The amount of the refraction varies with the apparent ze- 
nith distance. In the zenith it is zero, since the light passes per- 
pendicularly through all the strata of the atmosphere : and it is 
the greater, the greater is the zenith distance ; for, the greater the 
zenith distance of a star, the more obliquely does the light which 
comes from it to the eye penetrate the earth's atmosphere, and en- 
ter its different strata, and therefore, according to a well-known 
principle of optics, the greater is the refraction. 

94. To find the amount of the refraction for a given zenith dis 
tance or altitude. Let us first show a method of resolving this 
problem by tir^ ^eneral theory of refraction. According to this 
theory, the amount of the refraction, except so far as the convexity 
of the strata of the atmosphere may have an effect, depends whol- 
ly upon the absolute density of the air immediately in contact with 
the earth, and not at all upon the law of variation of the density of 
the different strata ; that is, the actual refraction is the same that 
would take place if the light passed from a vacuum immediately 

Yig. 23. hito a stratum of air of the 

density which obtains at the 
earth's surface. Let us sup 
pose, then, that the whole 
atmosphere is brought to the 
same density as that portion 
of it which is in contact witli 
the earth, and let bah (Fig. 
23) represent its surface ; 
also let O represent the sta- 
tion of the observer upon the 
earth's surface, and Sa a ray 
incident upon the atmosphere 
at a. Denote the angle of 
refraction OaC by p, and the 
refraction Oax by r. The 
angle of incidence 
Z'aS = Z'aS' + S'gS = OaC 

+ Oax =p + r. 
Now if we represent the in- 
dex of refraction of the atmosphere by m, we have, by the laws of 
refraction, 

sin Z'aS = m sin OaC, or sin (p + r) = m sin p ; 
developing (App. For. 15,) 

sin p cos r + cos p sin r = m sin p ; 
or, dividing by sin p, 

cos r + cot p sin r = m. 
But, as r is small, we may take cos r = 1, and sin r — r = r" sin 
1". (App. 47.) 

Whence, 1+ cot p .r" sin 1 " =m. or r" = — : — — - x =A tang p : 

* sin 1" cotp *> r ' 




REFRACTION. 



47 



putting A = 



m 



1 



Let ZCa = C ; and ZOa = Z. OaC 



sin 1" 

ZOa — ZCa, orp = Z — C. Substituting, we have r" = A tang 
(Z — C ;) or, omitting the double accent, and considering r as 
expressed in seconds, 

r = Atang(Z — C) (2) 

When the zenith distance is not great, C is very small with respect 
to Z. If we neglect it, we have 

r = A tang Z (3) ; 

which is the expression for the refraction, answering to the suppo- 
sition that the surface of the earth is a plane, ai. ■ that the light is 
transmitted through a stratum of uniformly dense air, parallel to 
its surface. We perceive, therefore, that the refraction, except in 
the vicinity of the horizon, varies nearly as the tangent of the ap- 
parent zenith distance. 

95. It has been ascertained by experiment that m, the index of 
refraction, (the barometer being = 29.6 inches, and the thermome- 
ter = 50°) = 1.0002803. Substituting in equation (3), after hav- 
ing restored the value of A, and reducing, there results 

r = 57".8 tang Z (4). 

96. With the aid of this formula, or of others purely theoretical, 
astronomers have sought to determine the precise amount of the 
refraction at various zenith distances from observation, and by col- 
lating the results of their observations to obtain empirical formulas 
that are more exact. 

97. One of the simplest methods of accomplishing this is the following: When 
the latitude or co-latitude of a place, and the polar distance of a star which passes 
the meridian near the zenith, have been determined, the refraction may be found 
for all altitudes from observation simply. 
For, let P (Fig. 24) be the elevated pole, 
Z the zenith, PZE the meridian, HOR 
the horizon, S the true place of a star, 
and S' its apparent place. Suppose the 
apparent zenith distance ZS' to have 
been measured. Now, in the triangle 
ZPS, ZP the co-latitude and PS the 
polar distance are known by hypothe- 
sis, and the angle P is the sidereal time 
which has elapsed since the star's last 
meridian transit, (or, if the star be to 
the east of the meridian, the difference 
between this interval and 24 sidereal 
hours,) converted into degrees by allow- 
ing 15° to the hour. Therefore we may 
compute the true zenith distance ZS, 
and subtracting from it the apparent 
zenith distance ZS', we shall have 
the refraction. For the solution of this problem the polar distance may be 
found by taking the complement of the declination computed from an ob- 
served meridian zenith distance, (68 ;) and, since the upper and lower transits 
of a circumpolar star take place at equal distances from the pole, the co-lati- 
tude may be found by taking the half sum of the greatest and least zenith dis- 
tances of the pole star. But it is obvious that neither of these quantities can be 
accurately determined, unless the measured zenith distances be corrected for re- 




48 CORRECTIONS OF CO-ORDINATES 

fraction. When, however, the zenith distances in question differ considerably from 
90°, the corresponding refractions may be at first ascertained with considerable 
accuracy by means of equation (4.) When more correct formulae have been ob- 
tained by this or any other process, the latitude and polar distance, and therefore 
the refraction answering to the measured zenith distance, will become more accu- 
rately known. 

98. The various formulae of refraction having been tested by 
numerous observations, it is found that they are all (ihough in dif- 
ferent degrees) liable to material errors, when the zenith distance 
exceeds 80°, or thereabouts. At greater zenith distances than this 
the refraction is irregular, or is frequently different in amount 
when the circumstances upon which it is supposed to depend are 
the same. 

99. The refractive power of the air varies with its density, and 
hence the refraction must vary with the height of the barometer 
and thermometer. 

100. The refractions which have place when the barometer 
stands at 29.6 inches, (or, according to some astronomers, 30 inch- 
es,) and the thermometer at 50°, are called mean refracti&ns. 

The refractions corresponding to any other height of the barom- 
eter or thermometer, are obtained by seeking the requisite correc- 
tions to be applied to the mean refractions, on the hypothesis that 
the refraction is directly proportional to the density of the atmo- 
sphere. 

101. To save astronomical observers and computers the trouble 
of calculating the refraction whenever it is needed, the mean re- 
fractions corresponding to various zenith distances or altitudes are 
computed from the formulae, as also the corrections for the barom- 
eter and thermometer, and inserted in a table. Table VIII is Dr. 
Young's table of mean refractions, and Table IX his table of cor- 
rections. The refraction answering to any zenith distance not 
inserted in the table can be found by simple proportion. (See 
Prob. VII.)* 

102. On inspecting Table VIII, it will be seen that the refrac- 
tion amounts to about 34' when a body is in the apparent horizon, 
and to about 58" when it has an altitude of 45°. 

OTHER EFFECTS OF ATMOSPHERICAL REFRACTION. 

103. Atmospherical refraction makes the apparent distance of 
any two heavenly bodies less than the true ; for it elevates them 
in vertical circles which continually approach each other from the 
horizon till they meet in the zenith. 

104. Refraction also makes the discs of the sun and moon ap- 
pear of an elliptical form when near the horizon. As it increases 
with an increase of zenith distance, the lower limb of the sun or 



* The tables referred to in the text maybe found near the end of the book. The 
problems referred to are in Part IV. 



PARALLAX. 49 

moon is more refracted than the upper, and thus the vertical diam- 
eter is shortened, while the horizontal diameter remains the same, 
or very nearly so. This effect is most observable near the hori- 
zon, for the reason that the increase of the refraction is there the 
most rapid. The difference between the vertical and horizontal 
diameters may amount to i part of the whole diameter. 

105. When a star appears to be in the horizon, it is actually 34' 
below it, (102 :) refraction, then, retards the setting and accele- 
rates the rising of the heavenly bodies. 

Having this effect upon the rising and setting of the sun, it must 
increase the length of the day. 

106. The apparent diameter of the sun is about 32' ; as this is 
less than the refraction in the horizon, it follows, that when the 
sun appears to touch the horizon it is actually entirely below it. 
The same is true of the moon, as its apparent diameter is nearly 
the same with that of the sun. 

PARALLAX. 

107. The correction for atmospherical refraction having been 
applied, the zenith distance of a body is reduced from the surface 
of the earth to its centre, by means of a correction called Parallax. 

108. Parallax is, in its most general sense, the angle made by 
the lines of direction, or the arc of the celes- 
tial sphere comprised between the places of 
an object, as viewed from two different sta- - 
tions. It may also be defined to be the an- 
gle subtended at an object by a line joining 
two different places of observation. Let S 
(Fig. 25) represent a celestial object, and 
A, B two places from which it is viewed. 
At A it will be referred to the point s of the 
celestial sphere, and at B to the point s' ; 
the angle BSA, or the arc ss' 9 is the paral- 
lax. The arc ss' is taken as the measure 
of the angle BSA, on the principle that the 
celestial sphere is a sphere of an indefinitely 
great radius, so that the point S is not sen- 
sibly removed from its centre. a b 

109. The term parallax is, however, generally used in astrono- 
my in a limited sense only, namely, to denote the angle included 
between the lines of direction of a heavenly body, as seen from a 
point on the earth's surface and from its centre ; or the angle sub- 
tended at a heavenly body by a radius of the earth. If C (Fig. 
26) is the centre of the earth, O a point on its surface, and S a 
heavenly body, OSC is the parallax of the body. 

110. The parallax of a heavenly body above the horizon is call- 
ed Parallax in Altitude. 

7 




50 



CORRECTIONS OF CO-ORDINATES. 



The parallax of a body at the time its apparent altitude is ze- 
ro, or when it is in the plane of the horizon is called the Horizon- 
tal Parallax of the body. Thus, if the body S (Fig. 26) be sup- 
Fig. 26. 




Fig. 27. 



posed to cross the plane of the horizon at S', OS'C will be its hori- 
zontal parallax. OSC is a parallax in altitude of this body, 

111. It is to be observed, that the definition just given of the hori- 
zontal parallax, answers to 
the supposition that the 
earth is of a spherical form. 
In point of fact, the earth 
(as will be shown in the se- 
quel) is a spheroid, and ac- 
cordingly the vertical and 
the radius at any point of 
its surface are inclined to 
each other ; as represented 
in Fig. 27, where OC is the 
radius, and OC the verti- 
cal. The points Z and z, 
in which the vertical and 
radius pierce the celestial 
sphere, are called, respec- 
tively, the Apparent Ze- 
nith and the True Zenith. 
In perfect strictness, the horizontal parallax is the parallax at the 
time zOS, the apparent distance from the true zenith, is 90°. 
No material error, however, will be committed in supposing the 




PARALLAX IN ALTITUDE, 51 

earth to be spherical, except when the question relates to the paral- 
lax of the moon. 

112. Let the apparent zenith distance ZOS=Z, (Fig. 26,) the 
true zenith distance ZCS —z y and the parallax OSC = p. Since 
the angle ZOS is the exterior angle of the triangle OSC, we have 

ZOS = ZCS + OSC, and hence also ZCS = ZOS — OSC ; 
or, 

Z = z -{-p, and z = Z — p . . . . (5). 

Thus, to obtain the trut zenith distance from the apparent, we have 
to subtract the parallax: and to obtain the apparent zenith distance 
from the true, to add the parallax. 

Parallax, then, takes effect wholly in a vertical plane, like the 
refraction, but in the inverse maimer ; depressing the star, while 
the refraction elevates it Thus, the refraction is added to Z, but 
the parallax is subtracted from it. 

113. To find, an expression for the parallax in altitude, 

(1.) In terms of the apparent zenith distance. — In the triangle 
SOC (Fig. 26) the angle OSC = parallax in altitude =p, OC =ra~ 
dius of the earth = R, CS = distance of the body S = D, and COS 
= 180° — ZOS = 180° — apparent zenith distance = 180° — Z j 
and we have by Trigonometry the proportion 

sin OSC : sin COS : : CO 4 CS ; 



whence, 
and 



or, 



smp : sin (180° — Z) :: R : D ; 
D smp =RsinZ ; 

R 

ship = j? sin Z .... . (6). 



D 

This equation shows that the parallax p depends for any given 
zenith distance Z upon the distance of the body, and is less in pro- 
portion as this distance is greater : also, that for any given distance 
of the body it increases with an increase in the zenith distance. 
When Z = 90°, p has its maximum value, and then = horizontal 
parallax = H ; and equa, (6) gives 

*inH = g (7): 

substituting, we have 

sinp = sin H sin Z .... (8). 

This last equation may be somewhat simplified. The distances of 
the heavenly bodies are so great, that p and H are always very 
small angles ; even for the moon, which is much the nearest, the 
value of H does not at any time exceed 62'. We may, therefore, 
without material error, replace smp and sin H hyp and H. This 
being done, there results, 

p = H sin Z . . . . (9), 



52 CORRECTIONS OF CO-ORDINATED 

Wherefore, the parallax in altitude equals the product of the hor- 
izontal parallax by the sine of the apparent zenith distance. 

If we take notice of the deviation of the earth's form from that 
of a sphere, Z, in equation (8), will represent the apparent distance 
from the true zenith, (111,) and H the horizontal parallax as it is 
denned in Art, 111, 

(2.) In terms of the true zenith distance. — In the actual state of astronomy* the 
true co-ordinates of the places of the heavenly bodies are generally known, or may 
be obtained by computation from the results of observations already made, and! 
from these there is often- occasion to deduce the apparent co-ordinates. For this 
purpose there is required an expression for the parallax in altitude in terms of tker 
true zenith distance. 

If we make Z = z -{-p (112) in equation (8), we shall have 

...... . „ sinp 

sin p = sm H sin Cz + p) r or sin H =■- f — - : 

r * r sin(z-r-jj) 



whence,. 

and 

Dividing, 



1 + mn H . 1 + ^Z-r = ±il±£l±^P r 
sin (z -p p) sm Qs + p) 

, . „ , ship sin(z + p) — sinp 

1 — SMI W = I : — - — ; '= : — ; ; - , 

sin (z -j- p) sm (z -f- p) 

1 -f- sin H sin (z-\-p)-\-smp w 
1 — sin H ~sin(z-\-p) — sinp' 



tang' (45<> + * H) = tan g (***+ £>, (see App. For. 36, 29) ; 

whence, 

tang (i*+?) = tang %z tang 2 (45°+iH) . . . (10). 
This equation makes known £ z + j>, from which we may obtain p by subtract- 
ing \ z. 

In order to be able to compute the parallax in altitude by means of 
formula (9) or ( 1 0), it is necessary to know H, the horizontal parallax. 

114. To find the horizontal parallax . 

Let O, O' (Fig. 27) represent two stations upon the same ter- 
restrial meridian OEO', and remote from each other, Z, Z' their 
apparent zeniths, and z, z' their true zeniths, QCE the equator, 
and S the body (supposed to be in the meridian) the parallax of 
which is to be found. Let the angle OSO' = A, zOS = Z, z'0'& 
= Z' ; also let CO = R, CO' = R', CS = D, the parallax in alti- 
tude OSC =p, and the parallax in altitude O'SC —p'. Now, by 
equation (6), replacing the sine of the parallax by the parallax it- 
self, (113,) 

p — T^sin Z, and p' = =r- sin Z' ; 

whence 

i_^ R • v , R' • ., RsinZ + R'sinZ'. 

P+y = D sm 5" s 15 * 



but, (equa. 7,; 



«. R ^ R 
H= s ,orD = H . 



HORIZONTAL PARALLAX. 53 

Substituting this value of D, and deducing the value of H, we 
have 

R(P+P') _ RxA 

R sin Z + R' sin Z' R sin Z + R' sin Z' K h 

It remains now to find an expression for A in terms of measura- 
ble quantities. Let Os and O's (Fig. 27) be the directions at 
and 0' of a fixed star which crosses the meridian nearly at the 
same time with the body- Owing to the immense distance of the 
star, these lines will be sensibly parallel to each other, (27.) Let 
the angle SOs, the difference between the meridian zenith dis- 
tances of the body and star, as observed at O, be represented by 
d, and let the same difference SO's for the station O', be represent- 
ed by d'. Now, 

OSO' = OLO'— SO's = SQ* — SO^, oxA=d — d'. 
If the body be seen on different sides of the star by the two ob- 
servers, we shall have 

A=d+d'. 
Substituting in equation (11), there results, 
H - *<d±d') 

RsinZ+R'sinZ' * * * K h 
If we regard the earth as a sphere, R = R', and dividing by R, 
we have 

H = -^|±* ...(13). 

smZ+srnZ' 

115. To find the parallax by means of these formulae, each of 
the two observers must measure the meridian zenith distance of 
the body, and also of a star which crosses the meridian nearly at 
the same time with the body, and correct them for refraction. The 
difference of the two will be 5 respectively, the values of d and d' ; 
and the corrected zenith distances of the body will be the values 
of Z and Z', if formula .(13) be used; if formula (12) be used, 
the measured zenith distances of the body must still be corrected 
for the reduction of latitude, (p. 19, Def. 4.) 

It is not necessary that the two stations should be on precisely 
the same meridian ; for if the meridian zenith distance of the body 
be observed from day to day, its daily variation will become 
known ; then, knowing also the difference of longitude of the two 
places, the following simple proportion will give the change of ze- 
nith distance during the interval of time employed by the body in 
moving from the meridian of the most easterly to that of the most 
westerly station, viz : as interval (T) of two successive transits : 
diff. of long., expressed in time, (t) : : variation of zenith dist. 
in interval T : its variation in interval t. This result, applied 
to the zenith distance observed at one of the stations, will re- 
duce it to what it would have been if the observation had been made 
in the same latitude on the meridian of the other station. 

116- The horizontal parallax of a heavenly body may be found 



54 CORRECTIONS OF CO-ORDINATES- 

by the foregoing process, to within 1" or 2" of the truth. No 
greater degree of accuracy is necessary in the case of the moon. 
But there are certain important uses made of the horizontal paral- 
lax of a body that will be noticed hereafter, which require that the 
parallax of the sun, and of the planets, should be known with much 
greater precision. The more accurate methods employed to deter- 
mine the parallaxes of these bodies will be explained (in principle 
at least) in subsequent parts of the work. 

117. In consequence of the spheroidal form of the earth, the hor- 
izontal parallax of a body is somewhat different at different places. 
Let H and H' denote the horizontal parallaxes of the same body, 
and R and R' the radii of the earth at two different places. Then,, 
by equation (7,) 

H=jj,andH'=jy; 

whence, 

H:H'::^:^::R:R'. 

Thus the parallax at the equator, called the Equatorial Paral- 
lax, is the greatest, and the parallax at the pole the least. The dif- 
ference between the parallaxes of the moon at the equator and at 
the pole may amount to about 12". For the other heavenly bodies 
the difference is too small to be taken into account. 

118. When the horizontal parallax has been found for any one 
distance and time from observation, the horizontal parallax for any 
other distance and time may be approximately computed, by means 
of the principle that the parallax of a body is directly proportional 
to its apparent diameter. The truth of this principle appears from 
the fact, that both the parallax (113) and the apparent diameter are 
inversely proportional to the same quantity, viz : the distance of 
the body from the earth. 

In the present condition of astronomical science, when the hori- 
zontal parallax of either one of the heavenly bodies is required for 
any particular time, it may.be obtained by computation, or from 
tables. It may also be taken out of the Nautical Almanac* 

119. The equatorial horizontal parallax of the moon varies from 
53' 48" to 61' 24", according to the distance of the moon from the 
earth. The equatorial parallax of the moon answering to the mean 
distance, is 57' 1". 

The horizontal parallax of the sun varies slightly, from a change 
of distance. At the mean distance it is 8" .6. 

The horizontal parallaxes of the planets are comprised within 
the limits 31", and 0".4. 

* The Nautical Almanac is a collection of data to be used in nautical and as 
tronomical calculations, published annually in England, and republished in New 
York. It may generally be obtained two or three years previous to the year for 
which it is calculated. 



ABERRATION. 55 

The fixed stars have no parallax.* 

120. Parallax in right ascension and declination, and in longi- 
tude and latitude. 

Since the parallax displaces a body in its vertical circle, which 
is generally oblique to the equator and ecliptic, it will alter 
its right ascension and declination, as well as its longitude and lat- 
itude. The difference between the true and apparent right ascen- 
sion is called the parallax in right ascension ; the like differences 
for the other co-ordinates are called, respectively, parallax in de- 
clination, parallax in longitude, and parallax in latitude. 

ABERRATION. 

121. The celebrated English astronomer, Dr. Bradley, com- 
menced in the year 1725 a series of accurate observations, with 
the view of ascertaining whether the apparent places of the fixed 
stars were subject to any direct alteration in consequence of the 
supposed continual change of the earth's position in space.. The 
observations showed that there had been in reality, during the pe- 
riod of observation, small changes in the apparent places of each 
of the stars observed, which, when greatest, amounted to about 
40" ; but they were not such as should have resulted from the sup- 
posed orbitual motion of the earth. These phenomena Dr. Brad- 
ley undertook to examine and reduce to a general law. After 
repeated trials, he at last succeeded in discovering their true ex- 
planation. His theory is, that they are different effects of one gen- 
eral cause, a progressive motion of light in conjunction with an 
orbitual motion of the earth. 

Fig. 28. 




A" 

122. Let us conceive the observer to be stationed at the earth's 
centre ; and let ACB (Fig. 28) be a portion of the earth's orbit, so 
small that it may be considered a right line, CS the true direction 

* The practical method of correcting for parallax is detailed and exemplified in 
Problem VIII. 



56 CORRECTIONS OF CO-ORDINATES. 

of a fixed star as seen from the point C, AC the distance through 
which the earth moves in some small portion of time, and aC the 
distance through which a particle of light moves in the same time. 
Then, a particle of light, which, coming from the star in the direc- 
tion SC, is at a at the same time that the earth is at A, will arrive at 
C at the same time that the earth does. Suppose that Aa is the 
position of the axis or central line of a telescope, when the earth 
is at A, and that, continuing parallel to itself, it takes up by virtue 

of the earth's motion, the successive positions A' a', k!'a" 

CS'. A particle of light which follows the line SC in space will 
descend along this axis : for aa' is to AA' and aa" is to AA", as 
aC is to AC, that is, as the velocity of light is to the velocity of 
the earth ; consequently, when the earth is at A' the particle of 
light is on the axis at a \ and when the earth is at A" the particle 
of light is on the axis at a", and so on for all the other positions of 
the axis, until the earth arrives at C. The apparent direction of 
the star S, as far, at least, as it depends upon the cause under con- 
sideration will therefore be CS'. 

The angle SCS', which expresses the change in the apparent 
place of a star S, produced by the motion of light combined with 
the motion of the spectator, is called the Aberration of the star ; 
and the phenomenon of the change of the apparent course of the 
light coming from a star, thus produced, is called Aberration of 
Light, or simply Aberration. 

123. The phenomenon of the aberration of light may be famil- 
iarly illustrated by taking falling drops of rain instead of particles 
of light, and a vessel in motion at sea instead of the earth moving 
through space ; and considering what direction must be given to 
a small tube by a person standing upon the deck of the vessel, so 
as to permit the drops falling perpendicularly to pass through the 
tube. It is plain, that if the tube had a precisely vertical position, its 
forward motion would bring the back part of the tube against the 
drop ; and that the only way to prevent this is to incline the upper 
end of the tube forward, or draw the lower end backward, whereby 
the back part of it would be made to pass through a greater dis- 
tance before it comes up to the line of descent of the drop. The 
quantity that it is made to deviate in direction from this line must 
depend upon the relative velocities of the falling drop and moving 
tube. To the observer, unconscious of his own motion, the drop 
will appear to fall in the oblique direction of the tube. 

124. If through the point a (Fig. 29) a line as' be drawn parallel 
to AC, and terminating in CS', the figure kas'C will be a parallel- 
ogram, and therefore as' will be equal to AC. Hence it appears, 
that if on CS, the line of direction of a star S, a line Ca be laid off, 
representing the velocity of light, and through a a line as' be drawn, 
having the same direction as the earth's motion and equal to its ve- 
locity, the line joining s' and C will be the apparent line of direc- 
tion of the star, the point S' its apparent place in the heavens > and- 



ABERRATION. 57 

the angle aCs' its aberration. We conclude, therefore, that by 
virtue of aberration a star is seen in advance of its true place, in the 
plane passing through the line of direction of the star and the !ine 
of the earth's motion. 

Fig. 29. 




The amount of the aberration of a star is always very small, 
(never greater than about 20",) because of the very great dispropor- 
tion between the velocity of light and the velocity of the earth. It 
is very much exaggerated in Figs. 28 and 29. 

125. The aberration is the same when a star is viewed with the 
naked eye, as when it is seen through a telescope. For, let aC, 
the velocity of the light, be decomposed into two velocities, of 
which one, AC, is equal and parallel to the velocity of the earth ; 
the other will be represented by s'C. Now, since the velocity 
AC is equal and parallel to the velocity of the earth, it will pro- 
duce no change in the relative position of a particle of light and 
the eye, and therefore the relative motion of the light and the eye 
will be the same that it would be if the earth were stationary and 
the light had only the velocity s'C ; accordingly, the light entering 
the eye just as it would do if it actually came in the direction s'C, 
and the eye were at rest, Cs' will be the apparent direction of the 
star from which it proceeds. 

126. If we regard the observer as situated upon the earth's 
surface, instead of being at its centre, the aberration resulting 
from the earth's motion of revolution will be still the same : for, 
all points of the earth advance at the same rate and in the same 
direction with the centre. The motion of rotation will produce an 
aberration proper to itself, but it is so small that there is no occa- 
sion to take it into account. 

127. To find a general expression for the aberration. — We have 
by Trigonometry, (Fig. 29,) 

sin AaC : sin CAa : : CA : Ca : : vel. of earth : vel. of light ; 
whence, 

sin AaC = sin CAa -~— , or, since AaC — SCS', 
Oa 

, . nk vel. of earth . 

sm aberr. = sm CAa ■ — - — , .. , . . . . (14). 
vel. of light v J 

When CAa is 90°, the aberration has its maximum value, and 

this has been found by observation to be about 20"(20".44) ; whence, 

8 



58 CORRECTIONS OF CO-ORDINATES. 

• «^,/ ve l- of earth ,- mS 

sm 20" = — = — 7-rr-rr . . . (15): 
vel. of light v J 

substituting, and taking sin BCa for sin CAa, to which it is very 
nearly equal, we. have 

sin aberr. = sin BCa sin 20" . . . (16). 
We may conclude from this equation, that the aberration in- 
creases with the angle BC« made by the direction of the star with 
the direction of the earth's motion ; that it is equal to zero when 
this angle is zero, and has its maximum value of 20" (more accu- 
rately 20" .44) when this angle is 90°. 

128. Let us now inquire into the entire effect of aberration in 
the course of a year. Let S (Fig. 30) be the sun ; E the earth ; 
E/g- its orbit ; ZTV that orbit extended to the fixed stars, or the 
ecliptic, (p. 15, Def. 17;) ET a tangent to the earth's orbit atE ; 
the place of S among the fixed stars or in the ecliptic, as seen 

from the earth ; s a fixed star ; 
sTY the arc of a great circle pass- 
ing through s and T. Then, by 
what has preceded, (124,) the 
earth moving in the direction 
E/g - , the apparent place of the 
star may be represented by s' and 
the aberration by sEis'. Thus, 
the effect of aberration at any one 
time is to displace the star by a 
small amount, directly towards 
the point T of the ecliptic, which 
is 90° behind the sun. As the 
earth moves, the position of the point T will vary ; and in the 
course of a year, while the earth describes its entire orbit in the 
direction Efg, this point will move in the same direction entirely 
around the ecliptic. In this period of time, therefore, ss' the small 
arc of aberration will revolve entirely around s the true position of 
the star ; from which we conclude, that in consequence of aberra- 
tion a star appears to describe a closed curve in the heavens around 
its true place. 

As the inclination of the direction of the star to the direction of 
the earth's motion will vary during a revolution of the earth, the 
aberration will also vary during this period, (127,) and hence the 
curve in question will not be a circle. It appears upon investiga- 
tion that it is an ellipse, having the true place of the star for its 
centre, and of which the semi-major axis is constant and equal to 
20". 44, and the semi-minor axis variable and expressed by 20". 44 
sin X, (X denoting the latitude of the star.) Each star, then, de- 
scribes an ellipse which is the more eccentric in proportion as the 
star is the nearer to the ecliptic ; for, the expression for the minor 
axis shows that the smaller the latitude the less will be this axis. 
For a star situated in the ecliptic the minor axis will be zero, and 




ABERRATION. 59 

the ellipse will be reduced to a right line. For a star in the pole 
of the ecliptic the minor axis is equal to the major, and the ellipse 
therefore becomes a circle. 

When the earth is at two diametrically opposite points of its orbit, as E and r, 
the direction of its motion, which is the same as that of the tangent to the orbit, 
will make equal angles with the line of direction of the star, but will be towards 
diametrically opposite points in the sphere of the heavens, (since the earth's orbit 
is to be considered as a mere point in the centre of this sphere, (27.) It follows, 
therefore, that in all such positions of the earth the aberration is the same, but in 
opposite directions. At E and r, where the angle sET included between the line 
of direction of the star and that of the earth's motion is 90°, the aberration is at 
its maximum, and the star is at the extremities of the major axis of its elliptic orbit. 
At / and g, 90° distant from E and r, this angle is at its minimum ; the aberration 
is the least possible, and the star is at the extremities of the minor axis of its orbit. 

129. Since aberration causes the apparent place of a star to dif- 
fer slightly from its true place, the true and apparent co-ordinates 
will, in consequence, differ somewhat from each other. The effects 
of the aberration of light upon the apparent right ascension and 
declination of a star, are called, respectively, the Aberration in 
Right Ascension and the Aberration in Declination. In like man- 
ner its effects upon the longitude and latitude are called the Aber- 
ration in Longitude and the Aberration in Latitude* 

130. Since the motion of the earth is at all times in a direction perpendicular, or 
nearly so, to the line followed by the light which comes from the sun to the earth, 
the aberration of the sun, which takes place only in longitude, is continually equal 
to 20".44, (127.) Thus, the sun's apparent place is always 20" behind its true place. 

131. For a planet, the aberration is different from what it is for a fixed star. 
As a planet changes its place during the time that the light is passing from it to 
the earth, it would, if the earth were stationary, appear to be as far behind its true 
place as it has moved during this interval. This aberration due to the motion of 
the planet, combined with that due to the earth's motion, will give the real aberra- 
tion of the planet. 

132. For the moon, the aberration occasioned by its motion around the earth is 
very small. The earth's motion produces no lunar aberration, for the reason that 
the moon, and consequently the light emitted from it, partakes of this motion. 

133. If the apparent places of a star, found at various times, be 
corrected for aberration, the same result for the true place of the 
star is obtained. Again, the deductions of Art. 128 agree in every 
particular with the observed phenomena of the apparent displace- 
ment of the stars, first discovered by Dr. Bradley. These facts 
show that the aberration of light is the true cause of these phe- 
nomena, and consequently, at the same time establish the fact of 
the earth's orbitual motion, as well as that of the progressive mo- 
tion of light. 

134. It may be worth while to state, that the first discovery of 
the progressive motion of light preceded the detection and expla- 
nation by Bradley of the phenomena of aberration. The discov- 
ery was made by Roemer, a Danish astronomer, in the year 1667, 
from a comparison of observations upon the eclipses of Jupiter's 
satellites. 

* For the practical method of determining and applying these corrections, see 
Probs. XIX., XXL, XXII., XXIII. 



60 



CORRECTIONS OF CO-ORDINATES. 



135. As to the actual velocity of light, we have, by equation 
(15,) vel. of earth : vel. of light : sin 20".44 : 1 : : 1 : 10,000, 
(nearly.) Taking the velocity of the earth at 68,167 miles per 
hour, and making the calculation by logarithms, we obtain for the 
velocity of light 191,000 (191,140) miles per second. As deter- 
mined from observations upon Jupiter's satellites, it is very nearly 
the same. The time employed by light in coming from the sun to 
the earth is 8m. 18s. 



PRECESSION AND NUTATION. 

136. In the investigations that follow, we shall take it for grant- 
ed that it is possible to find the obliquity of the ecliptic and the 
place of the equinox. Methods of determining them will be given 
when we come to treat of the apparent motion of the sun. 

137. By comparing the longitudes and latitudes of the same 
fixed stars, obtained at different periods, (69,) it is found that their 
latitudes continue very nearly the same, but that all their longi- 
tudes increase at the mean rate of about 50" per year. The longi- 
tude of a star being the arc of the ecliptic, intercepted in the order 
of the signs between the vernal equinox and a circle of latitude 
passing through the star, (p. 18, Def. 30,) it follows from the last 
mentioned circumstance, that the vernal equinox must have a mo- 
tion along the ecliptic in a direction contrary to the order of the 
signs, amounting to about 50" in a year. As it has been found 
that the autumnal equinox is always at the distance of 180° from 
the vernal, it must have the same motion. This retrograde motion 
of the equinoctial points, is called the Precession of the Equinoxes. 

138. As the latitude of a star is its distance from the ecliptic, 
(p. 18, Def. 31,) it follows from the circumstance of the latitudes 
of all the stars continuing very nearly the same, that the ecliptic 
remains fixed, or very nearly so, with respect to the situations of 
the fixed stars. 

139. The ecliptic being stationary, it is plain that the precession 

of the equinoxes mustresult from 
a continual slow motion of the 
equator in one direction. It ap- 
pears from observation, that the 
obliquity of the ecliptic, or the 
inclination of the equator to the 
ecliptic, remains, in the course of 
this motion, very nearly the same. 

140. Since the equator is in 
motion, its pole must change its 
place in the heavens. Let VLA 
(Fig. 31) represent the ecliptic, 
K its pole, which is stationary, 
P the position of the north pole 




PRECESSION. 61 

of the equator or of the heavens at any given time, and VEA the 
corresponding position of the line of the equinoxes : KPL re- 
presents the circle of latitude passing through P, or the solsti- 
tial colure. Now, the point V being at the same time in the 
ecliptic and equator, it is 90° distant from the two points K and P, 
the poles of these circles ; therefore, it is the pole of the circle 
KPL passing through these points, and hence VL = 90°. It fol- 
lows from this, that when the vernal equinox has retrograded to any 
point V, the pole of the equator, originally at P, will be found in 
the circle of latitude KP'L' for which V'L' equals 90° : it will 
also be at the distance KP' from the pole of the ecliptic, equal to 
KP. Whence it appears that the pole of the equator has a retro- 
grade motion in a small circle about the pole of the ecliptic, and 
at a distance from it equal to the obliquity of the ecliptic. As the 
motion of the equator which produces the precession of the equi- 
noxes is uniform, the motion of the pole must be uniform also ; 
and as the pole will accomplish a revolution in the same time with 
the equinox, its rate of motion must be the same as that of the 
equinox, that is, 50" of its circle in a year. The period of the 
revolution of the equinox and of the pole of the equator is some- 
thing less than 26,000 years. 

141. It is an interesting consequence of this motion of the pole 
of the equator and heavens, that the pole star, so called, will not 
always be nearer to the pole than any other star. The pole is at 
the present time approaching it, and it will continue to approach 
it until the present distance of 1£° becomes reduced to less than 
|°, which will happen about the year 2100 : after which it will 
begin to recede from it, and continue to recede, until about the year 
3200 another star will come to have the rank of a pole star. The 
motion of the pole still continuing, it will, in the lapse of centuries, 
pass in the vicinity of several pretty distinct stars in succession, 
and in about 13,000 years will be within a few degrees of the star 
Vega, in the constellation of the Lyre, the brightest star in the 
northern hemisphere. 

The present pole star has held that rank since the time of the 
celebrated astronomer Hipparchus, who flourished about 120 B.C. 
In very ancient times a pretty bright star in the constellation of 
the Dragon (a Draconis) was the pole star. 

142. The motion of the equator which produces the precession 
of the equinoxes, must also produce changes in the right ascensions 
and declinations of the stars. These changes will be different ac- 
cording to the situations of the stars with respect to the equator 
and equinoctial points. 

143. The ecliptic, although very nearly stationary, as stated in Art. 138, is not 
strictly so. By comparing the values of the obliquity of the ecliptic, found at dis- 
tant periods, it is ascertained that it is subject to a gradual diminution from century 
to century. A comparison of the results of observations made by Flamstead in 
1690, and by Dr. Maskelyne in 1769, gives for the mean secular diminution 50", 




62 CORRECTIONS OF CO-ORDINATES. 

and for the mean annual diminution 0".50. A more accurate determination of the 
mean annual diminution is 0".46. 

It appears from observation, that there are minute secular changes in the lati- 
tudes of the stars, which establish that the diminution of the obliquity of the eclip- 
tic arises from a slow displacement of the plane of the ecliptic (or of the earth's 
orbit) in space. 

144. If the ecliptic slowly changes its position in the heavens, its pole must like- 
wise ; and since the obliquity of the ecliptic is continually diminishing, its pole 
must be gradually approaching the pole of the equator. 

145. The motion of the ecliptic alters somewhat the precession of the equinoxes, 
making it a little less than it would be if the equator only was in motion : for, let 

•p- an EL (Fig. 32) represent the position of the eclip- 

tic, and VQ that of the equator, at any assumed 
date, and EL', VQ' the positions of the same 
circles at some later date ; the obliquity L'V'Q' 
at the second epoch being less than that (LVQ) 
at the first epoch : also let r be the physical point 
of the moveable ecliptic, which at the first epoch 
coincided with the point V, and n the point an- 
swering to V ; and Vr, Y'n the arcs of small 
circles described by the points V and V in the 
motion of the arc EVL about the point E. Since 
riV'V is a right angle, and Q'V'V an acute an- 
gle, the point n must fall to the left of V", and 
therefore V"r will be less than nr, or its equal 
VV', by the small arc wV". But VV is the 
precession on the fixed ecliptic, and rV" the ac- 
tual precession. We learn by the aid of Physi- 
cal Astronomy, that the amount of annual pre- 
cession would, if the ecliptic were fixed, be 50". 35. As we have already seen, the 
actual precession on the moveable ecliptic is 50", (more accurately, 50".23.) 

146. It remains for us now to take notice of a minute inequal- 
ity in the motion of the equator and its pole, which we have thus 
far overlooked. Dr. Bradley, in observing the polar distance of a 
certain star, (7 Draconis,) with the view of verifying his theory of 
aberration, discovered that the observed polar distance did not agree 
with the apparent polar distance, as computed from the results of 
previous observation, by allowing for precession, aberration, and 
refraction ; and hence inferred the existence of a new cause of vari- 
ation in the co-ordinates of a star. On continuing his observations, 
he found that the polar distance alternately increased and dimin- 
ished, and that it returned to the same value in about 19 years. 
These phenomena led him to suppose that the pole, instead of 
moving uniformly in a circle around the pole of the ecliptic, re- 
volved around a point conceived to move in this manner. 

If the pole has such a motion, it is plain that (allowing the fact 
of the earth's rotation) it must result from a vibratory motion of 
the earth's axis. To this supposed vibration of the axis of the 
earth, and consequently of that of the heavens, Dr.. Bradley gave 
the name of Nutation. The term Nutation is also applied to the 
changes of the co-ordinates of a star's place, which are produced 
by the nutation of the earth's axis. The point about which the 
pole was conceived to revolve, is the mean position of the pole, or 
the Mean Pole. 



NUTATION. 



63 



Dr. Bradley discovered, from his observations, that the curve described by the 
pole must be an ellipse, having its major axis in the solstitial colure ; and esti 
mated the value of the major axis at about 19", and that of the minor axis at about 
14". He also discovered that a connection existed between the position of the 
pole in its ellipse, and the position of the moon at the time its latitude was zero, 
(69,) and changing from south to north, or of the point in which the moon crossed 
the plane of the ecliptic in passing from the south to the north side of it, called the 



Fig. 33. 




ascending node of the moon's orbit ; for he 
found that the pole retrograded in like man- 
ner with the node ; that it completed its revo- 
lution in the same time, namely, in about 19 
years ; and that its position was determinable 
from the place of the node by a geometrical 
construction. Let P (Fig. 33) represent the 
mean pole, and p the true pole ; pfg' repre- 
sents the ellipse described by the true pole 
around P as a centre ; gg', lying in the sol- 
stitial colure KPL, being its major axis, and 
ff its minor axis. It is to be observed that 
the pole P is not stationary, but revolves in 
the circle NPP', carrying with it the ellipse 
pfg 1 . It will be seen that this ellipse is very 
much exaggerated in the figure : a true de- 
lineation of it on the scale of the figure would 
be altogether imperceptible. 

This theory of a nutation of the earth's axis has been verified by subsequent ob- 
servations, and Physical Astronomy has revealed the cause of the phenomenon. 

147. As the equator must move with the axis of the earth or heavens, nutation 
will change the position of the equinox and the obliquity of the ecliptic. It is plain 
that its effect upon the position of the equinox will be to make it oscillate periodi- 
cally and by equal degrees, from one side to the other of the position which corre- 
sponds to the mean pole ; and that its effect upon the obliquity of the ecliptic will 
be to make it alternately greater and less than the obliquity corresponding to the 
mean pole. The position of the equinox which corresponds to the mean pole, is 
called the Mean Equinox. The obliquity corresponding to the mean pole, is term- 
ed the Mean Obliquity. Mean Equator has a like signification. The real equinox 
and the real equator are called, respectively, the True Equinox and the True 
Equator. The actual obliquity of the ecliptic is termed the Apparent Obliquity. 
Right ascension and declination, as estimated from the true equator and true equi- 
nox, are called, respectively, True Right Ascension and True Declination ; and 
longitude, as reckoned from the true equinox, is called True Longitude. Right 
ascension, declination, and longitude, reckoned from the mean equinox and mean 
equator, are called, respectively, Mean Right Ascension, Mean Declination, and 
Mean Longitude. The true and mean co-ordinates differ by reason of nutation. 
The effect of nutation upon the right ascension is called the Nutation in Right 
Ascension; upon the declination, Nutation in Declination ; and upon the longi- 
tude, Nutation in Longitude. Its effect upon the obliquity of the ecliptic is called 
Nutation of Obliquity. The distance of the true from the mean equinox in longi- 
tude, which is the same as the nutation in longitude, is sometimes termed the 
Equation of the Equinoxes in Longitude ; and the distance in right ascension, the 
Equation of the Equinoxes in Right Ascension. The precession of the mean 
equinox is equal to the Mean Precession of the true equinox, which is 50" .2. 

148. Formulae for computing the nutation in right ascension, declination, &c, 
at any given time, are investigated in some astronomical works. These formulae 
cannot be used without a knowledge of the moon's motions. In practice, the nu- 
tations in right ascension, &c, are found by the aid of tables. (See Probs. XX., 
XXIII.) If these be applied to the true co-ordinates, the results will be the mean 
co-ordinates. If the mean co-ordinates be known, the same corrections will fur- 
nish the true. 

149. Physical Astronomy has made known the existence of another nutation of 
the earth's axis, too small to be detected by observation. It is called Solar Nuta* 
Hon. The nutation discovered by Dr. Bradley is frequently called Lunar Nutation 



64 



CORRECTIONS OF CO-ORDINATES. 



150. To reduce the co-ordinates of a star from one epoch to 
another. 

This problem is resolved by first converting the true co-ordi- 
nates into the mean, then transferring the mean co-ordinates from 
the one epoch to the other, and finally converting the reduced mean 
co-ordinates into the true. The mode of performing the first and 
last mentioned operations has already been considered, (148.) It 
remains now for us to show how to reduce mean co-ordinates from 
one epoch to another. 

(1.) When the interval of time between the epochs comprises 
but a few years. — In this case the changes, from precession, of 
the mean right ascension and declination in the course of a year, 
called the Annual Variation in right ascension and the Annual 
Variation in declination, are determined, then multiplied by the 
number of years in the interval, and applied as corrections to the 
given right ascension and declination. 

For this purpose formulae have been in- 
vestigated, in which the annual variations 
in right ascension and declination are ex- 
pressed in terms of the right ascension and 
declination of the star and the obliquity of 
the ecliptic. Let VLA (Fig. 34) be the 
ecliptic, K its pole, PP'P" the circle de- 
scribed by the mean pole, P the mean pole 
and VQA the mean equator at any given 
time, P' the mean pole and V'Q'A' the 
£/ mean equator a year afterwards, and s a 
star. Draw P'r perpendicular to the decli- 
nation circle Psa. We have 
an. var. in dec.=sa' — sa = Ps — T's = Pr ; 
but since PP'r may be considered as a right- 
angled plane triangle, 
Pr = PF cos FPr = PF sin QPa . . . . (17). 
Regarding KPF as a right-angled isosceles triangle, we obtain 
sin KPF or 1 : sin KF : : sin PKF : sin PF ; 
whence, 

sin PF = sin PKF sin KF, or PF = PKF sin KF (nearly) .... (18) : 
substituting in equation (17), there results, 

Pr = PKF sin KF sin QPa. 
PKF = 50".2 (140) ; KF = obliquity of the ecliptic = o> ; 
QPa = VQ — Vfl = 90° — R (R designating the right ascension of the star s.) 
Thus, finally, 

an. var. in dec. = 50".2 sin o> cos R . . . . (19). 
Next, we have 

an. var. in r. asc. = W — Va = W — mo = V'm -f- la' . . . . (20) ; 
but, 

V'm = VV cos VV'm = 50".2 cos o> ; 
and since the right-angled triangles sY'r and sba' are similar, 

sin sr or sin sP' (nearly) : sin P'r : : sin sa' : sin ha' ; 
whence, 

, sinsa' sin»o' 

sin ba = sin P'r . _. , or ha' = P'r . _, (nearly), 
sin F* sin F* v ' 

The triangle PP'r gives P'r = PF sin P'Pr = PF cos QPa = PKF sin KF cos 
QPa (equa. 18) ; and sin P's = cos sa'. Substituting, we obtain 

Jo' = PKF sin KF cos QPa ^^' = PKF sin KF cos QPa tang sa'. 
cos sa 




VARIATIONS OF THE CORRECTIONS. 65 

Replacing PKP', K.F, and QPa by their values, as above, and taking the declina- 
tion sa for sa' and denoting it by D, there results, 

ba' = 50".2 sin w sin R tang D. 
Now, substituting in equation (20) the values of Vm and ba', we have 

an. var. in r. ase. = 50".2 cos w + 50".2 sin u> sin R tang D . . . (21). 

The results of formulae (19, 21) are to be used with their algebraic signs, if the 
reduction is from an earlier to a later epoch, otherwise with the contrary signs. 
The declination is always to be considered positive if North, and negative if South. 

\'m = 50' .2 cos u = 50".2 cos 23° 28' = 46".0, 
is the annual retrograde motion of the equinoctial points along the equator. 

(2.) When the intci cal of the epochs is of considerable or great length. — If the 
epochs are separated by an interval of more than 10 or 12 years, the foregoing pn>- 
cess will not answer ; for in a period of ten years the annual variations will have 
sensibly altered.* In this case we may proceed as follows : Convert the right as- 
cension and declination into longitude and latitude, add to the longitude (or if the 
reduction be to an earlier epoch, subtract from it) the precession in longitude, 
which will be the product of 50". 23 by the interval of the epochs, expressed in years 
and parts of a year, and then with the longitude thus obtained, and the latitude, 
calculate the right ascension and declination, using the mean obliquity of the ecliptic. 

When the period is of great length, or very great precision is desired, the pre- 
cession on the fixed ecliptic should be used, which is 50". 35 per year, (145) ; ana 
the right ascension should be corrected for the change of the position of the equi- 
nox on the equator, produced by the motion of the ecliptic , which correction is 
— 0".13l3 (per year) for later epochs. 

REMARKS ON THE CORRECTIONS.— VERIFICATION OF THE 
HYPOTHESIS THAT THE DIURNAL MOTION OF THE STARS 
IS UNIFORM AND CIRCULAR. 

151. It appears from what we have stated on the subject of the 
Corrections : 1 . That Refraction varies during the day with the alti- 
tude of the body, and changes for all altitudes with the state of the 
atmosphere ; 2. That Parallax varies, like the refraction, with the 
altitude of the body, and changes from one day to another with its 
distance ; 3. That Aberration remains sensibly the same for two 
or three days, and depends for its absolute value on the time of the 
year ; 4. That Precession and Nutation do not perceptibly alter 
the co-ordinates of a star, unless it be a circumpolar star, under 
several days, and that the former increases uniformly with the time 
while the latter varies periodically, its effects entirely disappearing 
in about 19 years ; and, 5. That the absolute value of the Nutation 
depends entirely upon the longitude of the moon's ascending node .. 

152. In the determination of the amount and laws of the cor- 
rections, it was taken for granted by astronomers, that the diurnal 
motion of the stars was uniform and circular. This hypothesis 
may be verified in the following manner : Let the zenith distance 
and azimuth of the same star be measured at various times during 
a revolution, and corrected for refraction,, (the other corrections be 
ing insensible, (151.) ) Then, if the latitude of the place be 
known (68) in the triangle ZPS, (Fig. 17, p. 37,) we shall have ZP 

* It is to be understood that we are here giving methods of obtaining very accu- 
rate results. The process just explained, except for stars near the pole, will fur- 
nish results sufficiently accurate for most ourposes, even when the interval com- 
prises 20 years or more. 

9 



66 OP THE EARTH. 

the co-latitude, ZS the zenith distance of the star, and PZS its azi- 
muth, whence we may compute PS. If this calculation be made 
foi the time of each observation, it will be found that the same 
value for PS is obtained in every instance ; which proves the di- 
urnal motion to be circular. Again, let the angle ZPS be com- 
puted for the time of each observation, with the same data, and it 
will be found that it varies proportionally to the time ; which es- 
tablishes that the diurnal motion is also uniform, or, at least, sensi- 
bly so during one revolution. 

153. When the transits of a circumpolar star are observed at 
"intervals of several days, and allowance is made for the error of 
the rate of the clock, as determined from observations upon stars 
in the vicinity of the equator, and for the aberration in right ascen- 
sion, it is found that the sidereal times of the transits differ slightly 
from each other ; from which it appears that the diurnal motion of 
the stars is not strictly uniform. When, however, allowance is 
made for the precession and nutation in right ascension, this dif- 
ference disappears. We may hence conclude that the motion of 
rotation of the earth is uniform, and that the motions of the earth 
-and of its axis, which produce the phenomena of precession and 
nutation, alter the times of the transits of the stars, thereby making 
the period of the apparent revolution of a star to differ slightly 
from the period of the earth's rotation. 

It may be observed, that the greatest difference obtains in the 
case of the pole star, and is half a second. 



CHAPTER IV. 



OF THE EARTH ; ITS FIGURE AND DIMENSIONS '. LATITUDE AND 

LONGITUDE OF A PLACE. 

154. Although it is in general sufficient for astronomical pur- 
poses to regard the earth as a sphere, still it is necessary in some 
cases of astronomical observation and computation, when accurate 
results are desired, to take notice of its deviation from the spheri- 
cal form. No account need, however, be taken of the irregulari- 
ties of its surface, occasioned by mountains and valleys, as they 
are exceedingly minute when compared with the whole extent of 
the earth. It is to be understood, then, that by the figure of the earth 
is meant the general form of its surface, supposing it to be smooth, 
or that the surface of the land corresponded with that of the sea. 

155. The figure of the earth is ascertained from an examination 
of the form of the terrestrial meridians. 

A Degree of a terrestrial meridian is an arc of it corresponding 
to an inclination of 1° of the verticals at the extremities of the arc. 



FIGURE AND DIMENSIONS OF THE EARTH. 



67 




It is also called a Degree of Lat- Fi 35 

itude. Thus if QNE (Fig. 35) g * ' 

represent a terrestrial meridian, 

ab will be a degree of it if it be of 

such length that the angle aCb 

between the verticals Z'aC, Z6C, 

is 1°. 

156. The length of a degree 
at any place will serve as a meas- 
ure of the curvature of the me- 
ridian at that place ; for it is ob- 
vious, from considerations already- 
presented, (4,) that the earth, if 
not strictly spherical, must be 
nearly so, and therefore that a 
degree ab (Fig. 35) may, with 
but little if any error, be considered as an arc of 1° of a circle 
which has its centre at C, the point of intersection of the verticals 
C«, Cb, at the extremities of the arc. The curvature will then 
decrease in the same proportion as the radius of this circle in- 
creases, and therefore in the same proportion as the length of a 
degree increases. Wherefore, the form of a meridian may be de- 
termined by measuring the length of a degree at various latitudes. 

157. To determine the length of a degree of a terrestrial me- 
ridian. — To accomplish this, we have, 

(1.) To run a meridian line ; an operation which is performed 
in the following manner. An altitude and azimuth instrument (or 
some other instrument adapted to meridian observations) is first 
placed at the point of departure, and accurately adjusted to the 
meridian. A new station is then established by sighting forward 
with the telescope. To this station the instrument is removed, 
and is there adjusted to the meridian by sighting back to the first 
station. A third station is then established by sighting forward 
with the telescope as before, to which the instrument is removed. 
By thus continually establishing new stations, and carrying the 
instrument forward, the meridian line may be marked out for any 
required distance. The meridian adjustments may be corrected 
from time to time by astronomical observations, (51, 71.) 

(2.) To find the length of the arc passed over. — When the 
ground is level, the length of the arc may be directly measured. 
In case the nature of the ground is such as not to allow of a di- 
rect measurement, it may be calculated with equal precision, by 
means of a base line and a chain of triangles the angles of which 
are measured. 

(3,) To find the inclination of the verticals at the extreme sta- 
tions, — This angle may be obtained by measuring the meridian 
zenith distances of the same fixed star at the two stations, correct- 
ing them for refraction if they are observed about the same time, 



68 OP THE EARTH. 

and for refraction, aberration, precession, and nutation, if they are 
observed at different times, and taking their difference. For, let 
0, 0' (Fig. 35) be the two stations in question, Z, Z' their zeniths r 
and OS, O'S the directions of a fixed star, and we shall have 
OcO' = ZOI — OIc = ZOS — Z'lS = ZOS — Z'O'S ;. 
that is, the angle comprised between the verticals equal to the dif- 
ference of the meridian zenith distances of the same star. 

(4.) The length of an arc of the meridian, either somewhat 
greater or less than a degree, having been found by the foregoing 
operations, thence to compute the length of a degree. — Let N de- 
note the number of degrees and parts of a degree in the measured 
arc, A its length, and x the length of a degree. Then, allowing: 
that the earth for an extent of several degrees does not differ sen- 
sibly from a sphere, we may state the proportion 

1° x A 
N : A : : 1° ; x ; whence x = — == — . . .. (22). 

158. Degrees have been measured with the greatest possible 
care, at various latitudes and on various meridians. Upon a com- 
parison of the measured degrees, it appears that the length of a 
degree increases as we proceed from, the equator towards either 
pole. It follows, therefore, (156,) that the curvature of a meridian 
is greatest at the equator, and diminishes as we go towards the 
poles ; and consequently, that the earth is flattened at the poles. 

159. The fact of the decrease of the curvature of a terrestrial 
meridian from the equator to the poles, leads to the supposition 
that it is an ellipse, having its major axis in the plane of the equa- 
tor and its minor axis coincident with the axis of the earth. Ana- 
lytical investigations, founded on the lengths of a degree in differ- 
ent latitudes and on different meridians, have established that a 
meridian is, in fact, very nearly an ellipse, and that the earth has 
very nearly the form of an oblate spheroid. The same investiga- 
tions have also made known the dimensions of the earth. The 
amount of the oblateness at the poles is measured by the ratio of 
the difference of the equatorial and polar diameters to the equato- 
rial diameter, which is technically termed the Oblateness. 

160. The form of the earth has also been determined by other 
methods, which cannot here be explained. All the results, taken 

together, indicate an oblateness of ^rr-. 

The following are the dimensions of the earth in miles : 

Radius at the equator 3962,6 miles. 

Radius at the pole 3949.6 

Difference of equatorial and polar radii 13.0 

Mean radius, or at 45° latitude . . . 3956.1 

Mean length of a degree 69.05 

The fourth part of a meridian . . . 6214.2 

161. Owing to the elliptical form of a terrestrial meridian, the 



LATITUDE AND LONGITUDE OF A PLACE. 



69 



radius and vertical at a place do Fig. 36. 

not coincide. Let ENQS (Fig. 
36) represent a terrestrial me- 
ridian. For any point O situa- 
ted on this meridian, CO will be 
the radius, and the normal line 
ZON the vertical. The posi- 
tion of the vertical will always 
be such that the apparent zenith 
Z will lie between the true ze- 
nith z and the elevated pole P. 
The inclination of the radius to 
the vertical, or the angle CON, 
called -the reduction of latitude, is greatest at the latitude 45°, and 
is there equal to about 11|-'. 

162. The oblateness of the earth occasions some slight modifications in the 
effects of parallax, which are in some instances to be taken into account in com- 
puting the apparent azimuth and zenith distance of a body, from the known co- 
ordinates of its true place. 




DETERMINATION OF THE LATITUDE AND LONGITUDE OF 
A PLACE. 

163. The latitude and longitude of a place ascertain its situation 
upon the earth's surface, and are essential elements in many astro- 
nomical investigations, 

1 64. To find the latitude of a place. 

(1.) By the zenith distances or altitudes of a circumpolar star 
at its upper and lower transits, — The principle of this method has 
already been demonstrated, (68,) and shown to be a particular case 
of a well known principle of Fig. 37. 

arithmetical proportions ; the fol- 
lowing is a more complete proof 
of it. Let Z(Fig. 37) represent 
the zenith, HOR the horizon, P 
the pole, and S, S' the points at 
which the upper and lower tran- 
sits of a circumpolar star take 
place ; HP will be equal to the H 
latitude, (34,) and ZP will be equal to the co-latitude. Now, 
we have 

HP = HS + PS, and HP = HS' — PS' = HS' — PS ; 

whence, 2HP = HS + HS', or, HP = 

In like manner we obtain 

zs + zs 




HS + HS' 



.(23). 



ZP = 



. . (24). 



Wherefore, let the altitudes of a circumpolar star at its upper and 



70 OF THE EARTH. 

lowei transits be measured and corrected for refraction, and their 
half sum will be the latitude ; or, let the zenith distances be meas- 
ured, and corrected for refraction, and their half sum subtracted 
from 90° will be the latitude. Stars should be selected that have 
a considerable altitude at their inferior transit, for, the greater is 
the altitude the less is the uncertainty as to the amount of the 
refraction. On this principle the pole star is to be preferred to all 
others. 

(2.) By a single meridian altitude or zenith distance. — Let 
s, s', s" (Fig. 10, p. 20) be the points of meridian passage of three 
different stars, the first to the north of the zenith, the second be- 
tween the zenith and equator, and the third to the south of the 
equator : ZE = the latitude, and we have for the three stars, 

ZE = sE — Zs, ZE = s f E + Zs', ZE = Zs" — s»E. 

Thus, if the zenith distance be called north or south, according as 
the zenith is north or south of the star when on the meridian, in 
case the zenith distance and declination are of the same name 
their sum will be equal to the latitude ; but if they are of different 
names their difference will be the latitude, of the same name with 
the greater. 

This method supposes the declination of a body to be known. 
The declination of a star or of the sun at any time is, in practice r 
obtained for the solution of this and other problems, by the aid of 
tables, or is taken by inspection from the English Nautical Alma- 
nac, or other similar work. If the time of the meridian transit be 
known, the altitude may be measured by a sextant, (79). The ob- 
served altitude must be corrected for refraction, and also for paral- 
lax if the body observed is the sun, or moon, or either one of the 
planets. 

This method of finding the latitude is the one most generally 
employed at sea, the sun being the object observed. As the time 
of noon is not known with accuracy, several altitudes about the 
time of noon are taken, and the meridian altitude is deduced from 
these. 

165. The astronomical latitude being known, the reduced lati- 
tude (p. 19, Def. 4) may be obtained by subtracting from it the 
reduction of latitude. For, if OC (Fig. 36) represents the radius, 
and ON the vertical, at any place 0, and ECQ represents the ter- 
restrial equator, ONQ will be the astronomical latitude, OCQ the 
reduced latitude, and CON the reduction of latitude ; and we have 

ONQ = OCQ + CON, and OCQ = ONQ — CON . . (25). 
(For the practical method of resolving this problem, see Prob. XV.) 

166. There are various methods of finding the longitude of a 
place, nearly all of which rest upon the following principle : 

The difference at any instant between the local times, (whether 
sidereal or solar,) at any place and on the first meridian, is the 
longitude of the place, expressed in time ; and consequently, also p 



LONGITUDE OF A PLACE. 71 

the difference between the local times at any two places is their 
difference of longitude in time. 

The truth of this principle is easily established. In the first 
place, we remark that the longitude of a place contains the same 
number of degrees and parts of a degree as the arc of the celestial 
equator comprised between the meridian of Greenwich and the 
meridian of the place. Now, it is Oh. Om. Os. of mean solar time 
or mean noon at any place, when the mean sun (45) is on the me- 
ridian of that particular place. Therefore, as the mean sun, mov- 
ing in the equator, recedes from the meridian towards the west at 
the rate of 1 5° per mean solar hour, when it is mean noon at a 
place to the west of Greenwich, it will be as many hours and parts 
of an hour past mean noon at Greenwich, as is expressed by the 
quotient of the division of the arc of the celestial equator, or its 
equal the longitude, by 15. If the place be to the east, instead of 
to the west of Greenwich, when it is mean noon there it will be as 
much before mean noon at Greenwich as is expressed by the lon- 
gitude of the place converted into time, (as above.) In either situ- 
ation of the place, then, the principle just stated will be true. 

It is plain that the equality between the differences of the times 
and of the longitudes will subsist equally if sidereal instead of so- 
lar time be used. 

167. To find the longitude of a place.. 

(1.) Let two observers, stationed one at Greenwich and, the other 
at the given place, note the times of the occurrence of some phe- 
nomenon which is seen at the same instant at both places ; the 
difference of the observed times will be the longitude in time. 
These same observations made at any two places will make known 
their difference of longitude. If the stations are not distant from 
each other, a signal, as the flashing of gunpowder, or the firing of 
a rocket, may be observed. When they are remote from each other, 
celestial phenomena must be taken. Eclipses of the satellites of 
Jupiter and of the moon, are phenomena adapted to the purpose in 
question. However, as in these eclipses the diminution of the 
light of the body is not sudden, but gradual, the longitude cannot 
be obtained with very great accuracy from observations made upon 
them. 

(2,) Transport a chronometer which has been carefully adjust- 
ed to the local time at Greenwich, to the place whose longitude is 
sought, and compare the time given by the chronometer with the 
local time of the place. In the same way, by transporting a chro- 
nometer from any one place to another, their difference of longi- 
tude may be obtained. The error and rate of the chronometer 
must be determined at the outset, and as often afterwards as cir- 
cumstances will admit, that the error at the moment of the obser- 
vation may be known as accurately as possible. To ensure greater 
certainty and precision in the knowledge of the time, three or four 
chronometers are often taken, instead of one only. 



72 PLACES OF THE FIXED STARS. 

This method is much used at sea ; the local time being obtained 
from an observation upon the sun or some other heavenly body, in 
a manner to be hereafter explained. 

(3.) Let the Greenwich time of the occurrence of some celestial 
phenomenon be computed, and note the time of its occurrence at 
the given place. 

Eclipses of the sun and moon, and of Jupiter's satellites, occul- 
tations of the stars by the moon, and the angular distance of the 
moon from some one of the heavenly bodies, are the phenomena 
employed. The Greenwich times of the beginning and end of the 
eclipses of Jupiter's satellites, are published for the solution of 
the problem of the longitude in the English Nautical Almanac. 
Eclipses of the sun and occultations of the stars furnish the most 
exact determinations of the longitude, but they cannot be used 
for this purpose unless the longitude is already approximately 
known. 

The explanation, in detail, of the method of lunar distances, 
which is chiefly used at sea, may be found in treatises on Naviga- 
tion and Nautical Astronomy. 



CHAPTER V. 

OF THE PLACES OF THE FIXED STARS. 

168. The place of a fixed star in the sphere of the heavens is 
found by ascertaining its true right ascension and declination, which 
are the co-ordinates of its place. The process of finding the true 
right ascension and declination of a heavenly body has already 
been detailed : the apparent right ascension and declination are 
found as explained in Arts. 54, 68, and to these are applied the 
several corrections of refraction, parallax (when sensible,) and 
aberration, (92, 120, 129.) 

When right ascensions and declinations found at different times 
are to be compared together, or employed in the same calculations, 
as often becomes necessary, they are to be reduced to the same 
epoch by correcting for precession and nutation, (p. 64.) 

169. It is important to observe, however, that the places of the 
fixed stars, as at present known, were not obtained by the direct 
process just referred to, that is, by observing the right ascension 
and declination, and applying to them at once all the corrections 
of which we have treated. They were arrived at by successive 
approximations. The respective corrections were applied in suc- 
cession as they came to be discovered ; and more accurate results 
were obtained, as, by the improvement of the instruments, the ob- 



THE CONSTELLATIONS. 73 

serrations became more and more exact, and as the amount of the 
corrections came to be known with greater and greater precision. 

170. In order to distinguish the fixed stars from each other, they 
are arranged into groups, called Constellations, which are ima- 
gined to form the outlines of figures of men, animals, or other ob- 
jects, from which they are named. Thus, one group is conceived 
to form the figure of a Bear, another of a Lion, a third of a Dragon, 
and a fourth of a Lyre. The division of the stars into constella- 
tions is of very remote antiquity ; and the names given by the an- 
cients to individual constellations are still retained. 

The resemblance of the figure of a constellation to that of the 
animal or other object from which it is named, is in most instance^ 
altogether fanciful. Still, the prominent stars hold certain definite 
positions in the figure conceived to be drawn on the sphere of ihe 
heavens. Thus, the brightest star in the constellation Leo is placed 
in the heart of the Lion, and hence it has sometimes been called 
Cor Leonis, or the Lion's Heart : and the brightest star in the 
constellation Taurus is situated in the eye of the Bull, and there- 
fore sometimes called the BulVs Eye; while that conspicuous 
cluster of seven stars in this constellation, known by the name of 
the Pleiades, is placed in the neck of the figure. Again, the line 
of three bright stars noticed by every observer of the heavens in 
the beautiful constellation of Orion, is in the belt of the imaginary 
figure of this bold hunter drawn in the skies. The three larger 
stars of this constellation are, respectively, in the right shoulder, in 
the left shoulder, and in the left foot. 

171. The constellations are divided into three classes : North- 
ern Constellations, Southern Constellations, and Constellations of 
the Zodiac. Their whole number is 91 : Northern 34, Southern 
45, and Zodiacal 12. The number of the ancient constellations 
was but 48. The rest have been formed by modern astronomers 
from southern stars not visible to the ancient observers, and others 
variously situated, which escaped their notice, or were not atten- 
tively observed. 

172. The zodiacal constellations have the same names as the 
signs of the zodiac, (Def. 25, p. 17) : but it is important to observe 
that the individual signs and constellations do not occupy the same 
places in the heavens. The signs of the zodiac coincided with the 
zodiacal constellations of the same name, as now defined, about the 
year 140 B. C. Since then the equinoctial and solstitial points 
have retrograded nearly one sign : so that now the vernal equinox, 
or first point of the sign Aries, is near the beginning of the constel- 
lation Pisces ; the summer solstice, or first point of Cancer, near 
the beginning of the constellation Gemini ; the autumnal equinox, 
or first point of Libra, at the beginning of Virgo ; and the winter 
solstice, or first point of Capricornus, at the beginning of Sagittarius. 

It follows from this, that when the sun is in the sign Aries, he 
is in the constellation Pisces, and when in the sign Taurus, in the 

10 



74 PLACES OF THE FIXED STARS. 

constellation Aries, &c, &o. For the rest, it should T^e observed 
that the constellations and signs of the zodiac have not precisely 
the same extent. 

173. The stars of a constellation are distinguished from each 
other by the letters of the Greek alphabet, and in addition to these, 
if necessary, the Roman letters, and the numbers 1, 2, 3, &c. ; 
the characters, according to their order, denoting the relative mag- 
nitude of the stars. Thus, a Arietis designates the largest star in 
the constellation Aries ; {3 Draconis, the second star of the Drag- 
on, &c. 

Some of the fixed stars have particular names, as Sirius, Aide- 
bar an, Arcturus, Regulus, &c. 

174. The stars are also divided into classes, or magnitudes, ac- 
cording to the degrees of their apparent brightness. The largest 
or brightest are said to be of the first magnitude ; the next in order 
of brightness, of the second magnitude ; and so on to stars of the 
sixth magnitude, which includes all those that are barely percepti- 
ble to the naked eye. All of a smaller kind are called telescopic 
stars, being invisible without the assistance of the telescope. The 
classification according to apparent magnitude is continued with 
the telescopic stars down to stars of the twentieth magnitude, (ac- 
cording to Sir John Herschel,) and the twelfth according to Struve. 

The following are all the stars of the first magnitude that occur 
in the heavens, viz. Sirius, ox the Dog-star, Betelgeux, Rigel, Al- 
debaran, Capella, Procyon, Regulus, Denebola, Cor. Hydrce, 
Spica Virginis, Arcturus, Antares, Altair, Vega, Deneb or Alpha 
Cygni, Dubhe or Alpha Ursce Majoris, Alpherat or Alpha Andro- 
meda, Fomalhaut, Achernar, Canopus, Alpha Crucis, and Alpha 
Centauri. It is the practice of Astronomers to mark more or less 
of these stars as intermediate between the first and the second 
magnitude ; and in some catalogues some of them are assigned to 
the second magnitude. All of these stars, with the exception of 
the last four, come above the horizon in all parts of the United States. 

175. There are two principal modes of representing the stars ; 
the one by delineating them on a globe, where each star occupies 
the spot in which it would appear to an eye placed in the centre 
of the globe, and where the situations are reversed when we look 
down upon them ; the other is by a chart or map, where the stars 
are generally so arranged as to be represented in positions similar 
to their natural ones, or as they would appear on the internal con- 
cave surface of the globe.* The construction of a globe or chart 
is effected by means of the right ascensions and declinations of the 
stars. Two points diametrically opposite to each other on the 
surface of an artificial globe are taken to represent the poles of the 
heavens, and a circle traced 90° distant from these for the equator : 
another point 23^° from one of the poles is then fixed upon for one 

* Encyclopedia Metropolitana, Art. Astronomy, p. 505. 



RIGHT ASCENSION AND DECLINATION. 



75 



of the poles of the ecliptic, and with this point as a geometrical 
pole a great circle described ; the points of intersection of the 
two circles will represent the equinoctial points. The point which 
represents the place of a star is found by marking off the right as- 
cension and declination of the star upon the globe. 

All the fixed stars visible to the naked eye, together with some 
of the telescopic stars, are represented on celestial globes of 1 2 or 
18 inches in diameter. 

176. The places of the fixed stars are generally expressed by 
their right ascensions and declinations, but sometimes also by 
their longitudes and latitudes. A table containing a list of fixed 
stars designated by their proper characters, and giving their mean 
right ascensions and declinations, or their mean longitudes and lati- 
tudes, is called a Catalogue of those stars.* 

Table XC. is a catalogue of fifty principal fixed stars, and gives 
their mean right ascensions and declinations for the beginning of 
the year 1 840, as well as their annual variations in right ascension 
and declination. The annual variations serve to extend the use of 
the catalogue about 10 years (150) before and after the epoch for 
which it is constructed. (See Prob. XVIII.) Every ten years, or 
thereabouts, a new catalogue must be formed. 

177. If the true right ascension and declination of a star at a given time be re- 
quired, correct the mean right ascension and declination found by the catalogue, 
for nutation. (See Art. 148.) And if the apparent right ascension and declination 
be required, correct also for aberration. (See Art. 129.) 

178. The latitude and longitude of a fixed star or other heavenly 
body are obtained originally by computation from its right ascen- 
sion and declination. 

To convert the right ascension and declination of a body into 
its longitude and latitude. — Let EQ (Fig. 38) represent the equa- 
tor, EC the ecliptic, P, K the poles of 
the equator and ecliptic, E the vernal 
equinox, PSR a circle of declination 
and KSL a circle of latitude, both 
passing through a body S. The 
right ascension of the body is ER = 
R ; tlje declination RS = D ; the 
longitude EL = L ; and the latitude 
LS — X. REL = w is the obliqui- 
ty of the ecliptic, which is one of 
the essential data of the problem. 

* Various catalogues have at different periods been published. The first was be- 
gun by Hipparchus, 120 years before the Christian era. Of the modern catalogues, 
the following may be cited as among the most accurate, although not the most 
extensive, viz. the Catalogues of Flamstead, Lacaille, Bradley, Maskelyne, Piazzi, 
and of the Royal Astronomical Society, and of the British Association. 

The Nautical Almanac contains a Catalogue of 100 principal fixed stars, of 
which 54 are designated as Standard Stars — that is, stars whose places are sup. 
posed to be known with all attainable precision. The largest single catalogue ever 
published is the Histoire Celeste of Lalande, which gives the places of 50,0Q0 stars 




76 PLACES OF THE FIXED STARS. 

RES = x and LES = y are employed as auxiliary angles. In the 
right-angled spherical triangle LES we have by Napier's rules for 
the solution of right-angled triangles, (see Appendix,) 

sin (co. LES) = tang EL tang (co. ES) ; 
whence, 

tan EL = cos LES tan ES, or, tan L = cos (RES — w) tan ES ; 
but 

sin (co. RES) = tan ER tan (co. ES,) or, tan ES = tang T ^ ; 

cos KiLio 

thus, 

T /TI -n x tang ER cos (x — w) tan R , nn . 
ta nL = co.(RES- M )^g § =— L—J . . (26): 

and to find x, we have 

sin ER = tan (co. RES) tan RS, or, cot x = sin R cot D . . (27.) 

Again, 

sin EL = tan (co. LES) tan LS, or tan LS = tan LES sin EL, 

which gives 

tang X = tang (x — w) sin L . . . (28.) 

Equation (27) makes known the value of x, with which we de- 
rive the values of L and X by means of equations (26) and (28.) 
In resolving the equations attention must be paid to the signs of 
the quantities, which are determined according to the usual trigo- 
nometrical rules, it being understood that the declination D is to 
be regarded as negative when it is south, x is to be taken always 
less than 180°, and greater or less than 90° according as its cotan- 
gent is'negative or positive. L will always be in the same quad- 
rant with R. The latitude X will be north or south according as 
tang X comes out positive or negative. 

The apparent or mean obliquity is used, according as the case 
refers to true or mean co-ordinates. (For exemplifications of this 
problem see Prob. XXIV.) 

179. It is now frequently necessary to resolve the converse problem, that is, to 
convert the longitude and latitude of a body into its right ascension and decli- 
nation. 

The triangle RES (Fig. 38) gives 

sin (co. RES) = tang ER tang (co. ES) ; . # 

whence, 

tan ER = cos RES tan ES, or, tan R = cos (LES + «) tan ES ; 
but 

sin (co. LES) = tang EL tang (co. ES), or tan ES = taUg ; 

thus, 

■^ ,t™ , .tang EL cos (y -f- w) tang L 

tang R = cos (LES + w) 6 = ^_T_J &_ .... (29) . 

s v cos LES cos y v ■ 

and to find y, we have 

sin EL = tang (co. LES) tang LS, or cot y = sin L cot X . . (30). 
For tbe declination, we have 

sin ER = tan (co. RES) tan RS, or, tan RS = tan RES sin ER ; 
or, 

tang D = tang (y -f- w) sin R . . . (31.) * 



OBLIQUITY OF THE ECLIPTIC. T7 

The value of y being derived from equation (30) and substituted in equations 
(29) and (31), these equations will then make known the values of R and D. The 
signs of the quantities are determined by the usual trigonometrical rules, the lati- 
tude X being taken negative when south, y is always less than 180°, and greater 
or less than 90° according as its cotangent comes out negative or positive. R will 
be in the same quadrant as L. The declination will be north or south according 
as its tangent comes out positive or negative. (For exemplifications of this prob- 
lem see Prob. XXV.) 

180. Table XCII. contains the mean longitudes and latitudes 
of some of the principal fixed stars for the beginning of the year 
1840, together with their annual variations, which serve to make 
known the mean longitudes and latitudes at any other epoch. (See 
Prob. XVIII.) 

181. The fixed stars, so called, are not all of them, rigorously 
speaking, fixed or stationary in the heavens. It has been discov- 
ered that many of them have a very slow motion from year to year. 
These motions of the stars are called their Proper Motions. The 
annual variations in right ascension and declination, and in longi- 
tude and latitude, given in Tables XC. and XCII., are the varia- 
tions due both to the precession of the equinoxes and the proper 
motions of the stars. 



CHAPTER VI. 

OP THE APPARENT MOTION OF THE SUN IN THE HEAVENS. 

182. The sun's declination, and the difference of right ascension 
of the sun and some fixed star, found from day to day throughout 
a revolution, are the elements from which the circumstances of the 
sun's apparent motion are derived. 

The motion of the sun, as at present known, has been arrived at 
in the same approximative manner as the places of the fixed stars, 
(169.) It would, in fact, be theoretically impossible to correct the 
co-ordinates of the sun's apparent place for precession, nutation, 
and aberration, in the original determination of the sun's motion ; 
for, the knowledge of these corrections presupposes some know- 
ledge of the motion of the sun. 

183. The curve on the sphere of the heavens passing through 
the successive positions determined as above from day to day, is 
the ecliptic. If we suppose it to be a circle, as it appears to be, 
its position will result from the position of the equinoctial points 
and its obliquity to the equator. 

184. To find the obliquity of the ecliptic. — Let EQA (Fig. 39) 
represent the equator, EC A the ecliptic, and OC, OQ lines drawn 
through O the centre of the earth and perpendicular to AOE the 



78 



APPARENT MOTION OF THE SUN. 



line of the equinoxes ; then the angle COQ will be the obliquity 
of the ecliptic. This angle has for its measure the arc CQ, and 

therefore the obliquity of the eclip- 
tic is equal to the greatest decli- 
nation of the sun. It can but 
rarely happen that the time of the 
greatest declination will coincide 
with the instant of noon at the 
place where the observations are 
made, but it must fall within at 
9 least twelve hours of the noon for 
which the observed declination 
is the greatest. In this interval 
the change of declination cannot 
exceed 4", and therefore the greatest observed declination cannot 
differ more than 4" from the obliquity. A formula has been in- 
vestigated, which gives in terms of determinable quantities the 
difference between any of the greater declinations and the maxi- 
mum declination. By reducing by means of this formula a num- 
ber of the greater declinations to the maximum declination, and 
taking the mean of the individual results, a very accurate value of 
the obliquity may be found. 

185. To find the position of the vernal or autumnal equinox. 
(1.) On inspecting the observed declinations of the sun, it is seen 
that about the 21st of March the declination changes in the inter- 
val of two successive noons from south to north. The vernal 




Fig. 40. 




S:RS 



equinox occurs at some moment 
of this interval. Let RS, R'S' 
(Fig. 40) represent the declinations 
at the noons between which the 
equinox occurs : as one is north 
and the other south, their sum (S) 
will be the daily change of declina- 
tion at the time of the equinox. 
Denote the time from noon to noon 
by T. Now, to find the interval 
(x) between the noon preceding 
the equinox and the instant of the 
equinox, state the proportion 

T^RS. 
1 .*., g , 



on the principle that the declination changes for a day or more pro- 
portionally to the time. Next, take the daily change in right 
ascension (RR') on the day of the equinox and compute the value 
of RE, by the proportion 

T : x, or T x RS : : RR' : RE ; 



POSITION OF THE EQUINOX. 79 

add RE to MR, the observed difference of right ascension 
(182) on the day preceding the equinox, and the sum ME will be 
the distance of the equinox from the meridian of the star observed 
in connection with the sun.* 

The position of the autumnal equinox may be found by a simi- 
lar process, the only difference in the circumstances being that the 
declination changes from north to south instead of from south to 
north. 

If the value of x which results from the first proportion be add- 
ed to the time of noon on the day preceding the equinox, the result 
will be the time of the equinox. 

(2.) In the triangle RES (Fig. 39) we have the angle RES = u 
the obliquity of the ecliptic, and RS = D the declination of the 
sun, both of which we may suppose to be known, and we have by 
Napier's first rule, 

sin ER = tang (co. RES)tangRS = cot wtang D . . (32 ;) 
whence we can find ER. And by taking the sum or difference of 
ER and MR, according as the star observed is on the opposite 
side of the sun from the equinox or the same side, we obtain ME 
as before. If this calculation be effected for a number of posi- 
tions S, S', S", (fee, of the sun on different days, and a mean of 
all the individual results be taken, a more exact value of ME will 
be obtained. 

ME being accurately known, the precise time of the equinox 
may readily be deduced from the observed daily variation of right 
ascension on the day of the equinox. 

186. The calculations just mentioned rest upon the hypothesis 
that the ecliptic is a great circle. The close agreement which is 
found to subsist between the values of M E deduced from obser- 
vations upon the sun in different positions S, S', S", &c, estab- 
lishes the truth of this hypothesis. It is also confirmed by the 
fact, that the right ascensions of the vernal and autumnal equinox 
differ by 180°, since we may infer from this that the line of the 
equinoxes passes through the centre of the earth. 

187. The mean obliquity of the ecliptic is derived from the apparent obliquity, as 
well as the mean equinox from the true equinox, by correcting for nutation. 

188. The mean obliquity at any one epoch having been found, its value at any 
assumed time may be deduced from this by allowing for the annual diminution of 
0".46, (see Table XXII.) In like manner, the place of the mean equinox at any 
given time may be derived from its place once found, by allowing for the annual 
precession of 50". 23. 

The mean obliquity having thus been found for any assumed time, the apparent 
obliquity at the same time becomes known, by applying the nutation of obliquity. 
(SeeProb.X.) 

189. The longitude of the sun may be expressed in terms of 
the obliquity of the ecliptic and the right ascension or declination 
In the triangle ERS, (Fig. 39,) ES(=L) represents the longi- 

* The star is here supposed to be to the west of the sun. 



80 APPARENT MOTION OF THE STJN. 

tude of the sun supposed to be at S, ER (=R) its right ascension, 
and RS (=D) its declination. Now, by Napier's first rule, 

cosRES=tangERcotES,orcotES= COS -^ =cosREScotER; 
° tang ER 

thus, 

T -n T tangR . . 

cot L =cos w cot R, or tang L = — - — . . . (33). 

cos w 

Also, (Napier's second rule, Appendix,) 

sinRS =cos(co. RES)cos(co.ES),- whence, sin ES = . ^-e^ ; 

sm K-Ejo 

or, 

. T sinD , • . 

smL=- . . . (34). 

sm u 

With these formulae the longitude of the sun may be computed 
from either its right ascension or declination. (See Prob. XII.) 

Formulae (33) and (34) may be written thus, 

tang R = tang L cos w ; sin D = sin L sin u . . . (35). 

These formulae will make known the right ascension and decli- 
nation of the sun, when his longitude is given. (See Prob. XL) 
It will be seen in the sequel that in the present advanced state of 
astronomical science, the longitude of the sun at any assumed time 
may be computed from the ascertained laws and rate of the sun's 
motion. 

190. The interval between two successive returns of the sun to 
the same equinox, or to the same longitude, is called a Tropical 
Year. 

And the interval between two successive returns of the sun to 
the same position with respect to the fixed stars, is called a Side- 
real Year. 

191. It appears from observation that the length of the tropical 
year is subject to slight periodical variations. The period from 
which it deviates periodically and equally on both sides, is called 
the Mean Tropical Year. As the changes in the length, of the 
true tropical year are very minute, the length of the mean tropical 
year is obviously very nearly equal to the mean length of the true 
tropical year in an interval during which it passes one or more 
times through all its different values. In point of fact, it may be 
found with a very close approximation to the truth by comparing 
two equinoxes observed at an interval of 60 or 100 years. 

Theory shows that the variation in the length of the tropical year arises from 
the periodical inequality in the precession of the equinoxes which results from nu- 
tation, and certain periodical inequalities in the sun's yearly rate of motion ; and 
thus establishes also, that the mean tropical year, as above defined, is the same as 
the interval between two successive returns of the sun, supposed to have its mean 
motion, to the same mean equinox. 

According to the most accurate determinations, the length of the 
mean tropical year, expressed in mean solar time, is 365d. 5h. 
48m. 47.58s., (48s. nearly.) 



8i 

192. In a mean tropical year the sun's mean motion in longi- 
tude is 360° ; hence, to find his mean daily motion in longitude 
we have only to state the proportion 

365d. 5h. 48m. 48s. : Id. : : 360° : x = 59' 8".33. 

193. The sidereal year is longer than the tropical. — For since 
the equinox has a retrograde motion of 50". 23 in a year, when the 
sun has returned to the equinox it will not have accomplished a si- 
dereal revolution, into 50". 23. The excess of the sidereal over 
the tropical year results from the proportion 

59' 8".3 : 50".23 : : Id. : x = 20m. 23.1s. 
Thus the length of the mean sidereal year, expressed in mean 
solar time, is 365d. 6h. 9m. lis. 

194. If from the right ascensions and declinations of the sun, 
found on two successive days, the corresponding longitudes be de- 
duced (equas. 33, 34) and their difference taken, the result will be 
the sun's daily motion in longitude at the time of the observations. 
The sun's daily motion in longitude is not the same throughout 
the year, but, on the contrary, is continually varying. It gradually 
increases during one half of a revolution, and gradually decreases 
during the other half, and at the end of the year has recovered its 
original value. Thus, the greatest and least daily motions occur 
at opposite points of the ecliptic. They are, respectively, 61' 10" 
and 57' 11". 

195. The exact law of the sun's unequable motion can only be 
obtained by taking into account the variation of his distance from 
the earth ; for the two are essentially connected by the physical 
law of gravitation, which determines the nature of the earth's mo- 
tion of revolution around the sun. 

That the distance of the sun from the earth is in fact subject to 
a variation, may be inferred from the observed fact, that his ap- 
parent diameter varies. On measuring with the micrometer the 
apparent diameter of the sun from day to day throughout the year, 
it is found to be the greatest when the daily angular motion, or in 
longitude, is the greatest, and the least when the daily motion is 
the least ; and to vary gradually between these two limits. Ac- 
cordingly, the sun is nearest to us when his daily angular motion 
is the most rapid, and farthest from us when his daily motion is 
the slowest. The greatest apparent diameter of the sun is 32* 
36" ; and the least apparent diameter 31' 31". 

11 



82 



MOTIONS OF THE PLANETS IN SPACE. 



CHAPTER VII. 

OF THE MOTIONS OF THE SUN, MOON, AND PLANETS, IN 
THEIR ORBITS. 



KEPLER'S LAWS. 

196. The celebrated astronomer Kepler, who flourished early 
in the seventeenth century, by examining the observations upon 
the planets that had been made by the renowned Danish observer, 
Tycho Brahe, discovered that the motions of these bodies, and of 
the earth, were in conformity with the following laws : 

(1.) The areas described by the radius-vector of a planet [or 
the line drawn from the sun to the planet] are proportional to the 
times. 

(2.) The orbit of a planet is an ellipse, of which the sun occu- 
pies one of the foci. 

(3.) The squares of the times of revolution of the planets are 
proportional to the cubes of their mean distances from the sun, or 
of the semi-major axes of their orbits. 

These laws are known by the denomination of Kepler's Laws, 
They were announced by Kepler as the fundamental laws of the 
planetary motions, after a partial examination only of these mo- 
tions. They have since been completely verified by other astron- 
omers. We shall adopt the first two laws for the present as hy- 
potheses, and show in the sequel that they are verified by the 
results deducible from them. 

These laws being established, the third is obtained by simply 
comparing the known major axes and times of revolution. 

197. The apparent motion of the sun in space must be subject 
to Kepler's first two laws ; for the apparent orbit of the sun is of 
the same form and dimensions as the actual orbit of the earth, and 
the law and rate of the sun's motion in its apparent orbit, are the 
same as the law and rate of the earth's motion. To establish these 



Fig. 41. 




two facts, let EE'A (Fig. 
41) represent the elliptic or- 
bit of the earth, and S the 
position of the sun in space. 
If the earth move from E to 
any point E', as it seems to 
remain stationary at E, it is 
plain that the sun will ap- 
pear to move from S to a 
position S', on the line ES' 
drawn parallel to E'S the 
actual direction of the sun 
from the earth, and at a dis- 



LAW OF THE ANGULAR MOTION OF A PLANET. 83 

tance ES' equal to E'S the actual distance of the sun from the earth. 
Thus, for every position of the earth in its orbit, the corresponding 
apparent position of the sun is obtained by drawing a line parallel to 
the radius-vector of the earth, and equal to it. It follows, therefore, 
that the area SES' apparently described by the radius-vector of 
the sun (or the line drawn from the sun to the earth) in any inter- 
val of time, is equal to the area ESE' actually described by the 
radius-vector of the earth in the same time ; and consequently that 
the arc SS' apparently described by the sun in space, is equal to 
the arc EE 7 actually described in the same time by the earth. 
Whence we conclude, that the apparent motion of the sun in space, 
and the actual motion of the earth, are the same in every particular. 

198. It has been discovered that the motion of the moon in its 
revolution around the earth, is subject to the same laws as the mo- 
tion of a planet in its revolution around the sun. We shall assume 
this to be a fact, and show that our hypothesis is verified by the 
results to which it leads, 

199. That point of the orbit of a planet, which is nearest to the 
sun, is called the Perihelion, and that point which is most distant 
from the sun, the Aphelion^ The corresponding points of the 
moon's orbit, or of the sun's apparent orbit, are called, respective 
lv, the Perigee and the Apogee, 

These points are also called Apsides ; the former being termed 
the Lower Apsis, and the latter the Higher Apsis. The line join- 
ing them is denominated the Line of Apsides. 

The orbits of the sun, moon, and planets, being regarded as el- 
lipses, the perigee and apogee, or the perihelion and aphelion, are 
the extremities of the major axis of the orbit. 

200. The law of the angular motion of a planet about the sun 
may be deduced from Kepler's 
first law. Let PpAp" (Fig, 42) 
represent the orbit of a planet, con- 
sidered as an ellipse, and p, p' two 
positions of the planet at two in- 
stants separated by a short interval 
of time ; and let n be the middle 
point of the arc pp ! . With the ra- 
dius Sn describe the small circular 
arc Inl', and with the radius S6 
equal to unity describe the arc ab. 
It is plain that the two positions p,p' 
may be taken so near to each other, that the area Spp 1 will be 
sensibly equal to the circular sector SW. If we suppose this to 
be the case, as the measure of the sector is \lnV x Sn = \ab x 
Sn 2 , (substituting for Inl' its value ah x Sn,) we shall have 

area Spp' = \ah x Sn 2 . 
When the planet is at any other part of its orbit, as fl', if 




64 



MOTIONS OF THE PLANETS IN SPACE, 



Sp"p" be an area described in the same time as before, we shalE 
have 

area Sp'y ' = \a f b r x Srf 2 . 
But these areas are equal accordi ng to Kepler's first law r nence r 
iab x Sn = \a f V x Snf 2 . . . . (36) ; 
and ab : a r b' ; : S^ 2 : Sn, 

that is, the angular motion of a planet about the sun for a short 
interval of time, is inversely proportional to the square of the ra- 
dius-vector. 

It results from this that the angular motion is greatest at the pe- 
rihelion, and least at the aphelion, and the same at corresponding 
points on either side of the major axis; also, that it decreases pro- 
gressively from the perihelion to the aphelion, and increases pro- 
gressively from the aphelion to the perihelion, 

201. Now to compare the true with the mean angular motion, 
suppose a body to revolve in a circle around the sun, with the 
mean angular motion of a planet, and to set out at the same instant 
Fig. 43. with it from the perihelion. Let 

PMAM' (Fig. 43) represent the- 
elliptic orbit of the planet, and 
PBaB the circle described by the 
body. The position B of this fic- 
titious body at auy time will be the 
mean place of the planet as seen 
from the sun. The two bodies 
will accomplish a semi-revolution 
in the same period of time, and 
therefore be, respectively, at A and 
a at the same instant ; for it is ob- 
vious that the fictitious body will accomplish a semi-revolution in 
half the period of a whole revolution, and by Kepler's law of areas,, 
the planet will describe a semi-ellipse in half the time of a revolu- 
tion. At the outset, the motion of the planet is the most rapid, 
(200,) but it continually decreases until the planet reaches the 
aphelion, while the motion of the body remains constantly equal 
to the mean motion. The planet will therefore take the lead, and 
its angular distance pSB from the body will increase until its mo- 
tion becomes reduced to an equality with the mean motion, after 
which it will decrease until the planet has reached the aphelion A y 
where it will be zero. In the motion from the aphelion to the pe- 
rihelion, the angular velocity of the planet will at first be less than 
that of the body, (200,) but it will continually increase, while 
that of the body will remain unaltered : thus, the body will now 
get in advance of the planet, and their angular distance p'SB' will 
increase, as before, until the motion of the planet again attains to 
an equality with the mean motion, after which it will decrease, as 
before, until it again becomes zero at the perihelion. 




DEFINITIONS -OF TERMS. 85 

It appears, then, that from the perihelion to the aphelion the 
true place is in advance of the mean place, and that from the aphe- 
lion to the perihelion, on the contrary, the mean place is in ad- 
vance of the true place. 

The angular distance of the true place of a planet from its mean 
place, as it would be observed from the sun, is called the Equa- 
tion of the Centre. Thus, pSB is the equation of the centre cor- 
responding to the particular position p of the planet. It is evident, 
from the foregoing remarks, that the equation of the centre is zero 
at the perihelion and aphelion, and greatest at the two points, as 
M and M' ; , where the planet has its mean motion. The greatest 
value of the equation of the centre is called the Greatest Equation 
of the Centre. 

202. As the laws of the motion of the moon (198) and of the 
apparent motion of the sun (197) are the same as those of a planet, 
the principles established in the two preceding articles are as ap- 
plicable to these bodies in their revolution around the earth, as to 
a planet in its revolution around the sun. 

DEFINITIONS OF TERMS. 

263. (1.) The Geocentric Place of a body is its place as seen 
from the earth. 

(2.) The Heliocentric Place of a body is its place as it would 
be seen from the sun. 

(3.) Geocentric Longitude and Latitude appertain to the geo- 
centric place, and Heliocentric Longitude and Latitude to the he- 
liocentric place. 

(4.) Two heavenly bodies are said to be in Conjunction when 
their longitudes are the same, and to be in Opposition when their 
longitudes differ by 180°. When any one heavenly body is in 
conjunction with the sain, it is, for the sake of brevity, said to be 
in Conjunction ; and when it is in opposition to the sun, to be in 
Opposition. 

The planets Mercury and Venus, allowing that their distances 
from the sun are each less than the earth's distance (23), can never 
be in opposition. But they may be in conjunction, either by being 
between the sun and earth, or by being on the opposite side of the 
sun. In the former situation they are said to be in Inferior Con- 
junction, and in the latter in Superior Conjunction. 

(5.) A Synodic Revolution of a body is the interval between 
two consecutive conjunctions or oppositions. 

For the planets Mercury and Venus a synodic revolution is the 
interval between two consecutive inferior or superior conjunctions. 

(6.) The Periodic Time of a planet is the period of time in 
which it accomplishes a revolution around the sun. 

(7.) The Nodes of a planet's orbit, or of the moon's orbit, are 
the points in which the orbit cuts the plane of the ecliptic. The 



86 



MOTIONS OF THE PLANETS IN SPACE. 



node at which the planet passes from the south to the north side 
of the ecliptic is called the Ascending Node, and is designated by 
the character Q. The other is called the Descending Node, and 
is marked y. 

(8.) The Eccentricity of an elliptic orbit is the ratio which the 
distance between the centre of the orbit and either focus bears ta 
the semi-major axis. 

Fig. 44 




204. To illustrate these definitions, let EE'E" (Fig. 44) repre- 
sent the orbit of the earth ; CDC the orbit of Venus, or Mercury f 
which we will suppose, for the sake of simplicity, to lie in the 
plane of the ecliptic or of the earth's orbit ; LNP a part of the or- 
bit of Mars, or of any other planet more distant from the sun S 
than the earth is ; and ANB a part of the projection of this orbit 
on the plane of the ecliptic : N or Q, will represent the ascending 
node of the orbit ; and the descending node will be diametrically 
opposite to this in the direction Sn r . Also' let SV be the direction 
of the vernal equinox, as seen from the sun, and EV, E'V the par- 
allel directions of the same point, as seen from the earth in the two 
positions E and E' ; and P being supposed to be one position of 
Mars in his orbit, letp be the projection of that position on the 
plane of the ecliptic. The heliocentric longitude and latitude of 
Mars in the position P, are respectively VSp and P'Sp ; and if the 
earth be at E, his geocentric longitude and latitude are respec- 
tively VEp and PEp. If we suppose that when Mars is at P the 



ELEMENTS OF THE ORBIT OP A PLANET. 87 

earth is at E', he will be in conjunction ; and if we suppose the 
earth to be at E"' he will be in opposition. Again, if we suppose 
the earth to be at E, and Venus at C, she will be in superior con- 
junction ; but if we suppose that Venus is at C at the time that 
the earth is at E, she will be in inferior conjunction. The term 
inferior is used here in the sense of lower in place, or nearer the 
earth ; and. superior in the sense of higher in place, ox farther from 
the earth. Since the earth and planets are continually in motion, 
it is manifest that the positions of conjunction and opposition will 
recur at different parts of the orbit, and in process of time in every 
variety of position. The time employed by a planet in passing 
around from one position of conjunction, or opposition, to another, 
called the synodic revolution, is, for the same reason, longer than 
the periodic time, or time of passing around from one point of the 
orbit to the same again. 

ELEMENTS OF THE ORBIT OF A PLANET. 

205. To have a complete knowledge of the motions of the plan- 
ets, so as to be able to calculate the place of any one of them at 
any assumed time, it is necessary to know for each planet, in ad- 
dition to the laws of its motion discovered by Kepler, the position 
and dimensions of its orbit, its mean motion, and its place at a spe- 
cified epoch. These necessary particulars of information are sub- 
divided into seven distinct elements, called the Elements of the 
Orbit of a Planet, which are as follows : 

(1.) The longitude of the ascending node. 

(2.) The inclination of the plane of the orbit to the plane of the 
ecliptic, called the inclination of the orbit 

(3.) The mean distance of the planet from the sun, or the semi- 
major axis of its orbit. 

(4.) The eccentricity of the orbit 

(5.) The heliocentric longitude of the perihelion. 

(6.) The epoch of the planet being at its perihelion, or instead, 
its mean longitude at a given epoch. 

(7.) The periodic time of the planet. 

The first two ascertain the position of the plane of the planet's 
orbit ; the third and fourth, the dimensions of the orbit ; the fifth, 
the position of the orbit in its plane ; the sixth, the place of the 
planet at a given epoch ; and the seventh, its mean rate of motion. 

206. The elements of the earth's orbit, or of the sun's apparent 
orbit, are but^e in number ; the first two of the above -mentioned 
elements being wanting, as the plane of the orbit is coincident with 
the plane of the ecliptic. 

207. The elements of the moon's orbit are the same with those 
of a planet's orbit, it being understood that the perigee of the moon's 
orbit answers to the perihelion of a planet's orbit, and that the geo- 
centric longitude of the perigee and the geocentric longitude of the 



88 MOTIONS OF THE PLANETS IN SPACE. 

node of the moon's orbit answer, respectively, to the heliocentric 
longitude of the perihelion and the heliocentric longitude of the 
node of a planet's orbit. 

208. The linear unit adopted, in terms of which the semi-major 
axes, eccentricities, and radii-vectores of the planetary orbits, are 
expressed, is the mean distance of the sun from the earth, or the 
semi-major axis of the earth's orbit. When thus expressed, these 
lines are readily obtained in known measures whenever the mean 
distance of the sun becomes known. The lines of the moon's 
orbit are found in terms of the moon's mean distance from the 
earth, as unity. 

METHODS OF DETERMINING THE ELEMENTS OF THE SUN'S 
APPARENT ORBIT, OR OF THE EARTH'S REAL ORBIT. 

MEAN MOTION. 

209. The sun's mean daily motion in longitude results from the 
length of the mean tropical year obtained from observation, (192.) 

SEMI-MAJOR AXIS. 

210. As we have just stated, the semi-major axis of the sun's 
apparent orbit is the linear unit in terms of which the dimensions 
of the planetary orbits are expressed. Its absolute length is com- 
puted from the mean horizontal parallax of the sun. 

211. The horizontal parallax of a body being given, to find its 
distance from the earth, We have (equation 7, p. 51) 

sin H ' 
where H represents the horizontal parallax of the body, D its dis- 
tance from the centre of the earth, and R the radius of the earth. 
The parallax of all the heavenly bodies, with the exception of the 
moon, is so small, that it may, without material error, be taken in 
this equation in place of its sine. Thus, 

Again, since 6.2831853 is the length of the circumference of a 
circle of which the radius is 1, and 1296000 is the number of 
seconds in the circumference, we have 6.2831853 : 1 :: 1296000": 
x — 206264" .8 = the length of the radius (1) expressed in seconds. 
Hence, if the value of H be expressed in seconds, 

D = R 2 -^...(38). 
Jbi 

212. In the determination of the sun's parallax, by the process 
of Arts. 114 and 115, an error of 2" or 3", equal to about one- 
fourth of the whole parallax, may be committed, so that the dis- 
tance of the sun, as deduced by equation (38) from his parallax 
found in that manner, may be in error by an amount equal to one 



ECCENTRICITY OF THE SUN's APPARENT ORBIT. 89 

fourth or more of the true distance. There is a much more ac- 
curate method of obtaining the sun's parallax, which will be no- 
ticed hereafter. It has been found by the method to which we 
allude, that the horizontal parallax of the sun at the mean distance 
is 8".58, which may be relied upon as exact to within a small 
fraction of a second. We have, then, for the sun's mean distance, 
or the mean semi-major axis of his orbit, 

D = R 206264 - 8 = 24040.19 R = 95,102,992 miles ; 
8".58 
taking for R the mean radius of the earth = 3956 miles. 

ECCENTRICITY. 

213. First method. By the greatest and least daily motions 
in longitude. — We have already explained (194) the mode of de- 
riving from observation the sun's motion in longitude from day to 
day. Now, let v = the greatest daily motion in longitude ; v' = 
the least daily motion in longitude ; r = the least or perigean dis 
tance of the sun ; and r' the greatest or apogean distance ; and we 
shall have, by the principle of Art. 200, 

r : r' : : ^ v' : ^ v ; 

whence, r' + r : r' — r : : ^ v + *t v' : ^ v — v' v ', 

r' + r , J v + J v' ,— ,—. 
or, — ■ — : r' — r : : : v v — v v' : 

but, 

r' -f r 

= semi-major axis = 1 ; and r 1 — r = 2 (eccentricity) =2e; 

V v _|_ s/ v i — 
thus, 1 : 2e : : — : v v — v v', 

V i) — v^ v' 

and . e = ~-= —=r . . . (39). 

V V + V v' 

The greatest and least daily motions are, respectively, (at a 
mean,) 61 '.165 and 57'. 192. Substituting, we have 
e = 0.016791. 

The eccentricity may also be obtained from the greatest and 
least apparent diameters, by a process similar to the foregoing, on 
the principle that the distances of the sun at different times are in- 
versely proportional to his corresponding apparent diameters, (195.) 

214. Second method. By the greatest equation of the centre. 

(1.) To find the greatest equation of the centre. — Let L= the true longitude, 
and M = the mean longitude, at the time the true and mean motions are equal 
between the perigee and apogee, (201) ; L' = the true longitude andM' = th&mean 
longitude, when the motions are equal between the apogee and perigee ; and E = 
the greatest equation of the centre. Then (201) 

L = M + E, and L' = M' — E ; 
whence, L' — L = M' — M — 2E, 

„ d E = (M '- M) -< L '- L) ...(40). 

12 



90 MOTIONS OF THE PLANETS IN SPACE. 

About the time of the greatest equation the sun's true motion, and consequently 
the equation of the centre, continues very nearly the same for two or three days ; 
we may therefore, with but slight error, take the noon, when the sun is on either 
side of the line of apsides, that separates the two days on which the motions in 
longitude are most nearly equal to 59' 8", as the epoch of the greatest equation. 

The longitude Lor L' at either epoch thus ascertained, results from the observed 
right ascension and declination. M' — M = the mean motion in longitude in the 
interval of the epochs, and is found by multiplying the number of mean solar days 
and fractions of a day comprised in the interval, by 59' 8".330, the mean daily mo- 
tion in longitude. 

For example : from observations upon the sun, made by Dr. Maskelyne, in the 
year 1775, it is ascertained in the manner just explained that the sun was near its 
greatest equation at noon, or at Oh. 3m. 35s. mean solar time, on the 2d April, and 
at noon on the 31st, or at 23h. 49m. 35s. mean solar time, on the 30th of Septem- 
ber. The observed longitudes were, at the first period 12° 33' 39". 06, and at the 
second 188° 5' 44" .45. The interval of time between the two epochs is 182d. — 
14m. 

Mean motion in 182d. — 14m. . . . 179° 22' 41".56 
Difference of two longitudes .... 175 32 5 .39 



Difference 2 ) 3 50 36 .17 



Greatest equation of centre .... 1 55 18 .08 

More accurate results are obtained by reducing observations made during seve- 
ral days before and after the epoch of the greatest equation, and taking the mean 
of the different values of the greatest equation thus obtained. According to M. 
Delambre, the greatest equation was in 1775, 1° 55' 31".66. 

(2.) The eccentricity of an orbit may be derived from the greatest equation of 
the centre by means of the following formula : 

e 2 3.2 8 3.5.2>« C ' • - l '' 

in which K stands for the expression oqe;77qc; ^ being the greatest equation 

of the centre.) In the case of the sun's orbit, K being a small fraction, all its 
powers beyond the first may be omitted. Thus, retaining only the first term of the 
series, and taking E = 1° 55' 31".66 the greatest equation in 1775, we have 
K 1Q 55' 31".66 

6 = T = 2X570.2957795 = - 016803 - . 

215. It appears from the law of the angular velocity of a re- 
volving body, investigated in Art. 200, that the amount of the pro- 
portional variation of this velocity, which obtains in the course of 
a revolution, depends altogether upon the amount of the propor- 
tional variation of distance, or, in other words, upon the eccentri- 
city of the orbit, (Def. 8, p. 86.) It follows, therefore, that the 
amount of the greatest deviation of the true place from the mean 
place, that is, of the greatest equation of the centre, (201,) must 
depend upon the value of the eccentricity. If the eccentricity be 
great, the greatest equation of the centre will have a large value ; 
and if the eccentricity be equal to zero, that is, if the orbit be a 
circle, the equation of the centre will also be equal to zero, or the 
true and mean place will continually coincide. 

If either of the two quantities, the greatest equation and the 
eccentricity, be known, the other, then, will become determinate : 
and formulas have been investigated which make known either one 



91 

when the other is given. Equation 41 is the formula for the ec- 
centricity. 

• 216. From observations made at distant periods, it is discovered 
that the equation of the centre, and consequently the eccentricity, 
is subject to a continual slow diminution. The amount of the 
diminution of the greatest equation in a century, called the secular 
diminution, is 17". 2. 

LONGITUDE AND. EPOCH OF THE PERIGEE. 

217. As the sun's angular velocity is the greatest at the perigee, 
the longitude of the sun at the time its angular velocity is greatest, 
will be the longitude of the perigee. The time of the greatest 
angular velocity may easily be obtained within a few hours, by 
means of the daily motions in longitude, derived from observation. 

218. The more accurate method of determining the longitude 
and epoch of the perigee, rests upon the principle that the apogee 
and perigee are the only two points of the orbits whose longitudes 
differ by 180°, in passing from one to the other of which the sun 
employs just half a year. This principle may be inferred from 
Kepler's law of areas, for it is a well-known property of the ellipse, 
that the major axis is the only, line drawn through the focus that 
divides the ellipse into equal parts, and by the law in question 
equal areas correspond to equal times. 

219. By a comparison of the results of observations made at dis 
tant epochs, it is discovered that the longitude of the perigee is 
continually increasing at a mean rate of 61 ".5 per year. As the 
equinox retrogrades 50". 2 in a year, the perigee must then have a 
direct motion in space of 11". 3 per year. 

It will be seen, therefore, that the interval between the times of 
the sun's passage through the apogee and perigee, is not, strictly 
speaking, half a sidereal year, but exceeds this period by the inter 
val of time employed by the sun in moving through an arc of 5". 6 
the sidereal motion of the apogee and perigee in half a year. 

220. According to the most exact determinations, the mean Ion 
gitude of the perigee of the sun's orbit at the beginning of the yeai 
1800, was 279° 30' 8".39 : it is now 280£°. 

221. The heliocentric longitude of the perihelion of the earth's 
orbit, is equal to the geocentric longitude of the perigee of the sun's 
apparent orbit minus 180°. For, let AEP (Fig. 41, p. 82,) be the 
earth's orbit, and PV the direction of the vernal equinox. When 
the earth is in its perihelion P the sun is in its perigee S, and we 
have the heliocentric longitude of the perihelion VSP = VPL = 
angle abc — 180° = geocentric longitude of the sun's perigee — 
180°.* 

* It is plain that the same relation subsists between the heliocentric longitude 
of the earth and the geocentric longitude of the sun ia every other position of th€ 
earth in its orbit ; or that each point of the earth's orbit is diametrically opposite tc 
the corresponding point of the sun's apparent orbit. 



92 MOTIONS OF THE PLANETS IN SPACE. 

222. The epoch and mean longitude of the perigee of the sun's 
orbit being once found, the sun's mean longitude at any assumed 
epoch is easily obtained by means of the mean motion in longitude. 

METHODS OF DETERMINING THE ELEMENTS OF THE MOON'S 

ORBIT. 

LONGITUDE OF THE NODE. 

223. In order to obtain the longitude of the moon's ascending 
node, we have only to find the longitude of the moon at the time 
its latitude is zero and the moon is passing from the south to the 
north side of the ecliptic ; and this may be deduced from the lon- 
gitudes and latitudes of the moon, derived from observed right as- 
censions and declinations (69), by methods precisely analogous to 
those by which the right ascension of the sun, at the time its decli- 
nation is zero, and it is passing from the south to the north of the 
equator, or the position of the vernal equinox, is ascertained, (185.) 

INCLINATION OF THE ORBIT. 

224. Among the latitudes computed from the moon's observed 
right ascensions and declinations, the greatest measures the incli- 
nation of the orbit. It is found to be about 5° ; sometimes a little 
greater, and at other times a little less. 

MEAN MOTION. 

225. With the longitudes of the moon, found from day to day, 
it is easy to obtain the interval from the time at which the moon 
has any given longitude till it returns to the same longitude again. 
This interval is called a Tropical Revolution of the moon. It is 
found to be subject to considerable periodical variations, and thus 
one observed tropical revolution may differ materially from the 
mean period. In order to obtain the mean tropical revolution, we 
must compare two longitudes found at distant epochs. Their dif- 
ference, augmented by the product of 360° by the number of rev- 
olutions performed in the interval of the epochs, will be the mean 
motion in longitude in the interval, from which the mean motion in 
100 years or 36525 days, called the Secular motion, may be ob- 
tained by simple proportion. The secular motion being once 
known, it is easy to deduce from it the period in which the motion 
is 360°, which is the mean tropical revolution. 

It should be observed, however, that to find the precise mean secular motion in 
longitude, it is necessary to compare the mean longitudes instead of the true 
Now, the true longitude of the moon at any time having been found, the mean 
longitude at the same time is derived from it by correcting for the equation of the 
centre and certain other periodical inequalities of longitude hereafter to be noticed. 
But this cannot be done, even approximately, until the theory of the moon's mo- 
tions is known with more or less accuracy. 

226. The longitude of the moon, at certain epochs, maybe very 
conveniently deduced from observations upon lunar eclipses. For, 



MOON S MEAN MOTION IN LONGITUDE. 



93 



the time of the middle of the eclipse is very near the time of oppo- 
sition, when the longitude of the moon differs 180° from that of the 
sun, and the longitude of the sun results from the known theory 
of its motion. The recorded observations of the ancients upon the 
times of the occurrence of eclipses, are the only observations that 
can now be made use of for the direct determination of the longi- 
tude of the moon at an ancient epoch. 

227. The mean tropical revolution of the moon is found to be 

27.321582d. or 27d. 7h. 43m. 4.7s. (5s. nearly.) 
Hence, 27.321582d. : Id. : : 360° : 13°.17639. = 13° 10' 35" .0 = 
moon's mean daily motion in longitude. 

228. Since the equinox has a retrograde motion, the sidereal 
revolution of the moon must exceed the tropical revolution, as the 
sidereal year exceeds the tropical year. The excess will be equal 
to the time employed by the moon in describing the arc of preces- 
sion answering to a revolution of the moon. Thus, 

365.25d. : 50".2 : : 27.3d. : 3".75 = arc of precession, 
and 13°. 17 : Id. : : 3".75 : 6.8s. = excess. 

Wherefore, the mean sidereal revolution of the moon is 27d. 7h„ 
43m. 12s. 

229. It has been found, by determining the moon's mean rate of motion for pe- 
riods of various lengths, that it is subject to a continual slow acceleration. This 
acceleration will not, however, be indefinitely progressive : Laplace has investiga- 
ted its physical cause, and shown from the principles of Physical Astronomy, that 
it is really a periodical inequality in the moon's mean motion, which requires an 
immense length of time to go through its different values. 

The mean motion given in Art. 227 answers to the commencement of the pres- 
ent century. 

LONGITUDE OF THE PERIGEE, ECCENTRICITY, AND SEMI-MAJOR AXIS. 

230. The methods of determining these elements of the moon's 
orbit are similar to those by which the Fig. 45. 

corresponding elements of the sun's 
orbit are found. 

It is to be observed, however, that for the q 
longitudes of the sun, which are laid off in the 
plane of the ecliptic, in the case of the moon cor- 
responding angles are laid off in the plane of its 
orbit. These angles are reckoned from a line 
drawn making an angle with the line of nodes 
equal to the longitude of the ascending node, and 
are called Orbit Longitudes. The orbit longi- 
tude is equal to the moon's angular distance 
from the ascending node plus the longitude of 
the ascending node. Thus, let VNC (Fig. 45) 
represent the plane of the ecliptic, and V'NM a 
portion of the moon's orbit ; N being the as- 
cending node : also let EV be the direction of 
the vernal equinox, and let EV be drawn in the 
plane of the moon's orbit, making an angle 
V'EN with the line of the nodes equal to VEN, 
the longitude of the ascending node N. The 
orbit longitudes lie in the plane of the moon's orbit, and are estimated from this 
line, while the ecliptic longitudes lie in the plane of the ecliptic, and are estimated 




94 MOTIONS OP THE PLANETS IN SPACE. 

from the line EV. Thus, V'EM, or its measure V'NM, is the orbit longitude of 
the moon in the position M ; and VEra is the ecliptic longitude, that is, the longi- 
tude as it has been hitherto considered. V'NM = V'N -f NM = VN + NM ; that 
is, orbit long. = long, of Q + D's distance from Q. 

The orbit longitudes are calculated from the ecliptic longitudes ; these being de- 
rived from observed right ascensions and declinations. 

231. The ecliptic longitude of the moon at any time being given, to find the 
orbit longitude. — As we may suppose the longitude of the node to be given, (223) 
the equation of the preceding article will make known the orbit longitude so soon 
as MN, the moon's distance from the node, becomes known : now, by Napier's first 
rule, we have 

cos MNm = cot NM tang Nra ; 
or, cot NM = cos MNm cot Nm. 

Nm = ecliptic long. — long, of node ; and MNm = inclination of orbit. 

232. The horizontal parallax of the moon, like almost every 
other element of astronomical science, is subject to periodical 
changes of value. It varies not only during one revolution, but 
also from one revolution to another. The fixed and mean parallax 
about which the true parallax may be conceived to oscillate, an- 
swers to the mean distance, that is, the distance about which the 
true distance varies periodically, and is called the Constant of the 
Parallax. It is, for the equatorial radius of the earth, 57' 0".9 ,* 
from which we find by equation (38) the mean distance of the 
moon from the earth to be 60.3 radii, or about 240,000 miles. 

The first equation of article 211 would give a more accurate result. 
The greatest and least parallaxes of the moon are 61' 24" and 53' 48". 

233. The eccentricity of the moon's orbit is more than three 
times as great as that of the sun's orbit. Its greatest equation ex- 
ceeds 6° (215). 

MEAN LONGITUDE AT AN ASSIGNED EPOCH. 

234. We have already explained (225) the principle of the 
determination of the mean longitude of the moon from an ob- 
served true longitude. Now, when the mean longitude at any 
one epoch whatever becomes known, the mean longitude at any 
assigned epoch is easily deduced from it by means of the mean 
motion in longitude. 

METHODS OF DETERMINING THE ELEMENTS OF A PLANET'S 

ORBIT. 

235. The methods of determining the elements of the planetary 
orbits suppose the possibility of finding the heliocentric longitude 
and the radius-vector of the earth for any given time. Now, the 
elements of the earth's orbit having been found by the processes 
heretofore detailed, the longitude may be computed by means of 
Kepler's first law, and the radius-vector from the polar equation 
of the elliptic orbit. (See Davies' Analytical Geometry, p. 137.) 
The manner of effecting such computation will be considered 



LONGITUDE OF THE NODE OF A PLANET S ORBIT. 



95 



hereafter ; at present the possibility of effecting it will be taken for 
granted. 

HELIOCENTRIC LONGITUDE OF THE ASCENDING NODE. . 



236. When the planet is in either of its nodes, its latitude is zero. It follows, 
therefore, that the longitude of the planet at the time its latitude is zero, is the 
geocentric longitude of the node at the time the planet is passing through it. Now 
if the right ascension and declination of the planet be observed from day to day, 
about the time it is passing from one side of the ecliptic to the other, and convert- 
ed into longitude and latitude, the time at -pig-. 46 
which the latitude is zero, and the longitude 
at that time, may be obtained by a proportion. p. 
When the planet is again in the same node, 
the geocentric longitude of the node may 
again be found in the same manner as be- 
fore. On account of the different position c ' 
of the earth in its orbit, this longitude will 
differ from the former. 

Now, if two geocentric longitudes of the 
same node be found, its heliocentric longitude 
may be computed. — Let S (Fig. 46) be the 
sun, N the node, and E one of the positions 
of the earth for which the geocentric longi- 
tude of the node (VEN) is known. Denote 

this angle by G, the sun's longitude VES by I Sp\7 t -/•— 

S, and the radius-vector SE by r. Also, let \ I /\y / I 

E' be the other position of the earth, and \ 
denote the corresponding quantities for this 

position, VE'N, VE'S, and SE', respectively, — 

by G', S , and r'. Let the radius-vector of / 

the planet when in its node, or SN = V ; and 

the heliocentric longitude of the node, or VSN = X. The triangle SNE gives 




but 

and 

hence, 

or, 

In like manner 

Dividing, 

r sin (S — 



sin SNE : sin SEN : : SE : SN ; 

SEN = VES —VEN = S — G, 

SNE = VAN — VSN = VEN — VSN=G — X; 

sin (G — X): sin (S — G) : : r: V, 

rsin (S — G)=Vsin (G — X) . . . (42). 

r' sin (S' — G') = V sin (G — X.) 

rsin(S — G) sin (G — X) 



G) 



r' sin (S' - 
sin G cos X 



' r'sin (S' — GO 
whence, 

rsin (S 
r sin (S 



sin G' cos X — sin X cos G 



G') sin(G' — X)' 

sin X cos G sin G — cos G tang X 

~~ sin G' — cos G'tangX ' 



tang X 
Equation (42) gives 



G) sin G' — r' sin (S 7 — G') sin G 
G) cos G' — r' sin (S' — G') cos G 



• (43). 



r sin (S — G) 
sin (G — X) 



• • (44). 



237. The longitude of the node may also be found approximately 
from observations made upon the planet at the time of conjunction 
or opposition. It will happen in process of time that some of the 
conjunctions and oppositions will occur when the planet is near 
one of its nodes ; the observed longitude of the sun at this con- 
junction or opposition, will either be approximately the heliocentric 




96 MOTIONS OP THE PLANETS IN SPACE. 

longitude of the node in question, or will differ 180° from it. 
This will be seen on inspecting Fig. 47. If at a certain time the 

earth should be at E, crossing the 
line of nodes, and the planet in 
conjunction, it will be in the node 
N, and VES the longitude of the 
sun will be equal to VSN, the heli- 
ocentric longitude of the node. If 
the earth should be at E" and the 
planet in opposition, the longitude of 
the sun would be VE"S = VE"N 
+ 180° = VSN + 180° =hel. long, 
of node + 180°. 

If the daily variations of the lati- 
tude of the planet should be ob- 
served about the time of the sup- 
posed conjunction or opposition 
near the node, the time when the 
latitude becomes zero, or the pla- 
net is in its node, could approximately be calculated by simple 
proportion ; and then so soon as the rate of the angular motion 
about the sun becomes known (241) the longitude of the node 
could be more accurately determined. 

INCLINATION OF THE ORBIT. 

238. The longitude of the node having been found by the pre- 
ceding or some other method, compute the day on which the sun's 
longitude will be the same or nearly the same : the earth will then 
be on the line of the nodes. Observe on that day the planet's right 
ascension and declination, and deduce the geocentric longitude and 
latitude. Let ENp (Fig. 47) be the plane of the ecliptic, V the 
vernal equinox, S the sun, N the node, E the earth on the line of 
nodes, and P the planet as referred to the celestial sphere, from 
the earth. Let X denote the geocentric latitude Pp ; E the arc 
Np = Vp — VN = geo. long, of planet — long, of node ; and I 
the inclination PNp. The right-angled triangle PNp gives 

sin Np = tang Pp cot PNp = tang X cot I ; 

hence, cot I = -, and tang I = . °p . . . (45) : 

tangX' & smE v ' 

or, tang inclination = — : — -, ~ — '- — T — =-? . . . (46). 

sin (long. — long, of node) 

239. It will be understood, that to obtain an exact result, we must compute the 
precise time of the day at which the longitude of the sun is the same as that of 
the node, and then, by means of their observed daily variations, correct the longi- 
tude and latitude of the planet for the variations in the interval between the time 
thus ascertained and the time of the observation above mentioned. 



REDUCTION OF OBSERVATIONS. 97 

PERIODIC TIME. 

240. The interval from the time the planet is in one of its nodes 
till its return to the same, gives the periodic time or sidereal revo- 
lution. 

241. Another and more accurate method is to observe the length 
of a synodic revolution, (p. 85,) and compute the periodic time 
from this. If we compare the time of a conjunction which has 
been observed in modern times, with that of a conjunction observed 
by the earlier astronomers, and divide the interval between them 
by the number of synodic revolutions contained in it, we shall 
have the mean synodic revolution with great exactness, from which 
the mean periodic time may be deduced.* 

The periodic time being known, the mean daily motion around 
the sun may be found by dividing 360° by the periodic time ex- 
pressed in days and parts of a day. 

TO FIND THE HELIOCENTRIC LONGITUDE AND LATITUDE, AND THE 
RADIUS-VECTOR, FOR A GIVEN TIME. 

242. The earth being in constant motion in its orbit, and being 
thus at different times very differently situated with regard to the 
other planets, as well in respect to distance as direction, it is ne- 
cessary for the purpose of comparing the observations made upon 
these bodies with each other, to refer them all to one common 
point of observation. As the sun is the fixed centre about which 
the revolutions of the planets are performed, it is the point best 
suited to this purpose, and accordingly it is to the sun that the 
observations are in reality referred. The reduction of observations 
from the earth to the sun, as it is actually performed, consists in 
the deduction of the heliocentric longitude and latitude from the 
geocentric longitude and latitude, these being derived from the 
observed right ascension and declination. 

We will now show how to effect this deduction, supposing that the longitude of 
the node and the inclination of the orbit are known. Let NP (Fig. 48) be part of 
the orbit of a planet, SNC the plane of the ecliptic, N the ascending node, S the 
sun, E the earth, and P the planet ; also, let Pt be a perpendicular let fall from P 
upon the plane of the ecliptic, and EV, SV, the direction of the vernal equinox. 
Let X = PE?r the geocentric latitude of the planet ; I = PStt its heliocentric lati- 
tude ; G = VEtt its geocentric longitude ; L = VS* its heliocentric longitude ; 
S = VES the longitude of the sun ; N = VSN the heliocentric longitude of the 
node; I = PNC the inclination of the orbit; r = SE the radius-vector of the 
earth ; and v = SP the radius-vector of the planet. 

The point x is called the reduced place of the planet, and S^ its curtate distance. 
All the angles of the triangle SEtt have also received particular appellations : SttE 
the angle subtended at the reduced place of the planet by the radius of the earth's 
orbit, is called the Annual Parallax, SEtt the Elongation, and EStt the Commu- 

* We shall, in the sequel, investigate the equation that expresses the relation be- 
tween the synodic revolution and the periodic time. (See equation 129, p. 187) : if 
the synodic revolution (s) be given, then, the sidereal year (P) being also known, 
the value of the sidereal revolution of the planet (p) can be calculated from this 
equation. 

13 



98 MOTIONS OP THE PLANETS IN SPACE. 

Fig. 48. 




tation. Let A = SirE, E = SEjr, and C =EStt. Draw Stt' parallel to Ejt 
then A=irS:r'=:VS7r — VS:r' = VStf — VEtt=L— G; E=VE7r— VES = 

G _ S ; C = VSE — VSt: = 180° + VSE' — VS ff = 180° + VES — VStt = 180° 

4-S — L = T — L (putting T =1800 + S). 

(1.) For the latitude. — The triangles EPtt, SPtt, give 

t. ■, tang X Sir 

En- tang A = Pjt = Stt tang /, whence ■ — — ■ = =r- ; 
6 & tang I En ' 

^ . „ . ^, Sff sin E 

S?r : Ett : : sin E : sin C, or, =- = - — 77 ; 
Ett sin C 

tang A __ sin E 
tang I sin C ' 



but, 



substituting 
whence, 



(47). 



tang X sin C = tang I sin E . . (46) ; 
or, tang X sin (T — L) = tang I sin (G — S) . . 

Again, the triangle NPp gives, by Napier's first rule, 

sin Np = cot PNp tan Pp, or, sin (L — N) = cot I tan I . . (48). 
Either of the equations (47) and (48) will give the value of I, when the longi- 
tude L is known. 

(2.) For the longitude. — If we substitute in equation (47) the value of tang /, 
given by equation (48), and replace (G — S) by E, we have 

tang X sin (T — L) = sin (L — N) tang I sin E ; 
but T — L = (T — N) — (L — N) = D — (L—N), (denoting (T — N) by D) ; 
substituting, and designating L — N by x, 

tang X sin (D — x) = sin x tang I sin E ; 
whence, 

tang X sin D cos x — tang X cos D sin x = tang I sin E sin x, 
or, tang X sin D — tang X cos D tang x = tang I sin E tang x, 

which gives 

tang X sin D 
tang x 



tang X cos D + tang I sin E 
Substituting tho values of x, D, and E, we have, finally, 

tangX sin (T — N) 



(49.) 



tang(L— N) = 



(50). 



tang X cos (T — N) -f tang I sin (G — S) 
AsN is known, the value of L will result from this equation. 

243. The co-01 linates employed to fix the position of a planet 
in the plane of its orbit, are its orbit longitude (230) and its radius- 
vector, both of which result from the heliocentric longitude and 



REDUCTION OF OBSERVATIONS. 



99 



•atitude, the longitude of the node and the inclination of the orbit 
being known. 

In Fig. 48, V'NP represents the orbit longitude, and SP (= v) the radius-vector, 
for the position P. Now, the triangle PS* gives 

C T> S* Sff 

SP= — -,or,t; = 7 ; 

cos PSt cos I 



and the triangle ES* gives 

sin A : sin E : : SE : S* = 

whence, by substitution, 

r sin E 



SE sin E r sin E 



sin A 



sin A 



rsin(G — S) 



~~ sin A cos I sin (L — G) cos I 
The orbit longitude L' = NP + long. of node . . . (52). 
And to find NP, the triangle NPj9 gives 

tang N^ 



(51). 



cos PN^J = cot NP tang Np, or tang NP = 



cos I 



• (52); 



and Np = long, of planet — long, of node . 

244. The heliocentric longitude 
may be obtained in a very simple 
manner, if the observations be made 
upon the planet at the time of con- 
junction or opposition ; for, it will 
then either be equal to the geocen- 
tric longitude, or differ 1 80° from it. 

When the heliocentric longitude is 
thus found, the latitude for the same 
time may be obtained by solving the 
triangle PNp, (Fig. 49.) For, by Na- 
pier's first rule, 

sin Np == cot PNp tang Pp, 

or tang Pp = sin Np tang PNp ; 
where Pp is the latitude sought, 
PNp the known inclination of the orbit, and Np = VNp — VN = 
long, of planet — long, of node, both of which may be considered 
as known. 

The radius-vector may be computed for the same time from 
the triangle ESP ; for the side SE, the radius-vector of the earth, 
is known, as well as the angle SEP the geocentric latitude of the 
planet, and the angle ESP = 180° — PSp =180° — heliocentric lat. 

245. The radius-vector of either of the inferior planets at the 
time of maximum elongation, or greatest angular distance from the 




sun, may be approximately 
deduced from the amount of 
the maximum elongation, de- 
termined from observation. 
The elongation which obtains 
at any time may be found by 
ascertaining from instrumental 
observations the places of the 



Fig. 50. 




100 



MOTIONS OP THE PLANETS IN SPACE. 



Fig. 51. 



planet and sun in the heavens, and connecting these by an arc of a 
great circle, and with the pole by other arcs. In the triangle 
PSp (Fig. 50) thus formed there will be known the two polar dis- 
tances PS and Pp, which are the complements of the observed de- 
clinations, and the angle SPp the difference of their observed right 
ascensions, from which the angular distance Sp between the two 

bodies may be calculated. The 
maximum elongation being,, 
then, supposed to be known, 
let NPP' (Fig. 51) represent 
the orbit of an inferior planet. 
The line EP drawn from the 
earth to the planet will, at the 
time of maximum elongation,, 
be perpendicular to SP the 
radius-vector of the planet, 
and thus we shall have in the 
right-angled triangle EPS, the 
line ES, and the angle SEP, 
from which the radius-vector 
SP may be computed. 

As the earth and planet are 
in motion, the greatest elongation will occur at different points of 
the planet's orbit, and therefore we may find by the foregoing pro- 
cess different radii-vectores. 

LONGITUDE OF THE PERIHELION, ECCENTRICITY, AND SEMI-MAJOR 

AXIS. 




Fig. 52. 



246. The longitude of the perihelion, 
the eccentricity, and the semi-major 
axis, may be derived from the helio- 
centric orbit longitude (243) and the 
radius-vector found for three different 
times. 

Let SP, SP', SP" (Fig. 52) be the 
three given radii-vectores, V'SP, V'SP', 
V'SP", the three given longitudes, and 
AB the line of apsides of the planet's 
orbit. Let the angles PSF, PSP" P 
which are known, be represented by 
m, n, and the angle BSP, which is 
' unknown, by x ; and let the three ra- 
dii-vectores SP, SF, SP", be denoted by v, v', v" ; the semi-major axis AC by 
a, and the eccentricity by e : then, the three unknown quantities which are to be 
determined, are a, e, and the angle x, and the general polar equation of the ellipse 
furnishes for their determination the three equations 

"e-* ...(53), 




f = 



«"*= 



1 4- € COS X 

a(l — e 2 ) 
1 + « cos (x + m) 

a(i-e') 

l-t-«cos (x+n) 



(54), 



(55). 



101 

Equating the values of a (1 — c 2 ) obtained from equations (53) and (54), we have 
v-\-ve cos x = v' -f- v'e cos (x -j- wi), 

. . . (56). 
, . . (57* 



v cos x — v' cos (x -J- m) 
In like manner from (53) and (55), 

v" — v 



v cos x — v" ces (x -j- w) 
Let c' — v =p, and »" — v = q; then, by equating the second members of 
equations (56), (57), and transforming, we obtain 

p v cos x — v 1 cos (x -f- ?n) 

9 c cos a; — v" cos (a; -f n) 

v cos a: — u' cos m cos x -{- v ' sm w* sin x 
t> cos x — v" ces to cos x + t/' sin n sin a: 
c — »' cos m+u' sin m tang x 
v — v" cos n-\-v" sin n tang x 
p r v — v " C os to) — a (v — c'eoswi) ,_ . 

whence, tang x = — — ^— ■ « - - C 58 *- 

50' sm m — j)c" sin to 

The value of x being found by this equation, and subtracted from the orbit lon- 
gitude of the planet in the first position P, the result will be the orbit longitude of 
the perihelion. Also, x being known, e may be computed from either of the equa- 
tions (56) and (57) : and hence again, the semi-major axis from equation (53), 
(54), or (55). 

247. The semi-major axis or mean distance from the sun, may 
also be had by taking the mean of a great number of radii-vectores 
found for every variety of position of the planet in its orbit, (244), 
(245). 

248. Now that Kepler's third law has been established by in- 
vestigations in Physical Astronomy, it furnishes the most accurate- 
method of finding the mean distance of a planet from the sun. 
Thus, let P = the periodic time of the planet, and a = its mean 
distance ; then, the length of the sidereal year being 365.256374 
days, (193). 

(365.256374d.) 2 :P 2 ;:l 3 :a 3 ; 

249. If a great number of radii-vectores in a great variety of po- 
sitions of the planet in its orbit be found by the method explained 
in Art. 244, the longitude of the planet at the time it has the least 
calculated radius-vector will be approximately the longitude of the 
perihelion : or, if it chances that among the radii-vectores deter- 
mined there are two equal to each other, the position of the line 
of apsides may be found by bisecting the angle included between 
these. The ratio of the difference between the greatest and least 
calculated radii-vectores to the mean of the whole, will be the ap- 
proximate value of the eccentricity. 

EPOCH OF A PLANET BEING AT THE PERIHELION OF ITS ORBIT. 

250. From several observations upon the planet, about the time 



102 MOTIONS OF THE PLANETS IN SPACE. 

it has the same longitude as the perihelion, the correct time of its 
being at the perihelion may be easily determined by proportion. 

251. The mean longitude at an assigned epoch is obtained up- 
on the same principles as the mean longitude of the sun or moon, 
(222, 234.) 

REMARKS. 

252. The foregoing methods of determining the elements of a 
planet's orbit suppose observations to be made at two or more 
successive returns of the planet to its node : but it is not necessary 
to wait for the passage of a planet through its node. Soon after 
the planet Uranus was discovered by Sir William Hersehel, La- 
place contrived methods by which the elements of its elliptic orbit 
were determined from four observations within little more than a 
year from its first discovery by Hersehel.* After the discovery of 
Ceres, Gauss invented another general method of calculating the 
orbit of a planet from three observations, and applied it to the de- 
termination of the orbit of Ceres, and, subsequently, to the deter- 
mination of the orbits of Pallas, Juno, and Vesta. This method 
can be more readily employed in practice than that of Laplace, or 
than any of the solutions which other mathematicians have given 
of the same problem, and is now generally used by astronomers. 

MEAN ELEMENTS AND THEIR VARIATIONS. 

253. The elements of the planetary orbits, obtained by the foregoing' processes, 
are the true elements at the periods when the observations are made. Upon deter- 
mining them at different periods, it appears that they are subject to minute varia- 
tions. A comparison of the values found at various distant epochs shows that they 
are slowly changing from century to century, and that the changes experienced 
during equal long periods of time are very nearly the same. The amount of the 
variation of an element in a period of 100 years is called its Secular Variation. 
Upon reducing the elements, found at different times, to the same epochs by allow- 
ing for the proportional parts of the secular variations, the different results for each 
element are found to differ slightly from each other, which shows that the elements 
are also subject to slight periodical variations. These variations being very minute, 
the true elements can never differ much from the mean, or those from which they 
deviate periodically and equally on both sides. 

The mean elements at an assigned epoch may be had by finding the true ele- 
ments at various times, and reducing them to the given epoch, by making allow- 
ance for the proportional parts of the secular variations, and then taking for each 
element the mean of all the particular values obtained for it. 

254. A comparison of the mean values of the same element, found at distant 
epochs, makes known the variation of its mean value in the interval between 
them, from which the secular variation may be deduced by simple proportion. 

255. The elements of the moon's orbit are also subject to continual variations. 
These are, for the most part, periodic, and are far greater than the variations of 
the corresponding elements of a planet's orbit. It will be seen, then, that in de- 
termining the mean elements, a much greater number of observations will be 
required than in the case of a planetary orbit. The mean node and perigee have 
a rapid and nearly uniform progressive motion. Theory shows that the other 
mean elements, with the exception of the semi-major axis, are subject to secular 
variations, but their effect has hitherto been very inconsiderable. 

* History of the Inductive Sciences, vol. ii. p. 231. 



ELEMENTS OF THE PLANETARY ORBITS. 103 

256. The mean elements, which have been derived as above directly from ob- 
servation, have subsequently been verified and corrected, by comparing the com- 
puted with the observed places of the planet ; and for this purpose many thousands 
of observations have been made. 

257. Tables II. and III. contain the elements of the orbits of 
the principal planets, and of the moon's orbit, together with their 
secular variations, for the beginning of the year 1801 ; and also, 
the elements of the orbits of the four small planets, Vesta, Juno, 
Ceres, and Pallas, for 1831. (See Note III.) 

If an element be desired for any time different from the epoch 
of the table, we have only to allow for the proportional part of the 
secular variation, in the interval between the given time and the 
epoch of the table. 

258. It will be seen, on inspecting Table II., that the mean 
distances of the planets from the sun, or the semi-major axes of 
their orbits, are the only elements that are invariable. The rest 
are subject to minute secular variations. The nodes have all 
retrograde motions. The perihelia, on the contrary, have direct 
motions, with the single exception of the perihelion qf the orbit 
of Venus, which has a retrograde motion. The eccentricities of 
some of the orbits are increasing, of others diminishing. That 
of the earth's orbit is diminishing. 

The node of the moon's orbit has a retrograde motion, and the 
perihelion a direct motion. The former accomplishes a tropical 
revolution in 6788.50982 days, or about 18 years 214 days ; and 
the latter in 3231.4751 days, or in about 8 years 309 days. The 
mean motion of the node, and the mean motion of the perigee, are 
both subject to a slow secular diminution. 

259. It will be seen, also, that the orbits of the planets are 
ellipses of small eccentricity, or which differ but slightly from 
circles ; and that they are, with the exception of the orbit of 
Pallas, inclined under small angles to the plane of the ecliptic. 
The eccentricity is in every instance so small, that if a represen- 
tation of the orbit were accurately delineated, it would not differ 
perceptibly from a circle. The most eccentric orbits, among those 
of the seven principal planets, are those of Mercury and Mars ; 
and the least eccentric, those of Venus and the earth. The eccen- 
tricity of Mercury's orbit is 12 times that of the earth's, of Mars' 
6 times, of Venus' \. The eccentricities of the orbits of Jupiter, 
Saturn, and Uranus, are each about 3 times greater than that of 
the earth's orbit. 

The orbit of Mercury is more inclined to the ecliptic than the 
orbit of any other of the seven principal planets ; and the orbit of 
Uranus is less inclined than that of any other planet. The in- 
clination of the latter is f °, of the former 7°. 

The orbits of the four asteroids are more eccentric, and more 
inclined to the plane of the ecliptic, than those of the other planets 
in general. 



104 MOTIONS OP THE PLANETS IN SPACE. 

260. The mean distances of the planets from the sun are, m 
round numbers, as follows : Mercury 37 millions of miles, Venus 
69 millions of miles, the earth 95 millions of miles, Mars 145 mil- 
lions of miles, Juno 254 millions of miles, Jupiter 495 millions of 
miles, Saturn 907 millions of miles, Uranus 1824 millions of miles. 
The range of distance is from 1 to 77. The distance of Uranus 
is about 19 times the earth's distance: of Neptune 30 times. 

261. The approximate periods of revolution of the planets are 
as follows : Mercury 3 months, (I of a year,) Venus 1\ months, 
(f of a year,) Mars If years, Juno 4f years, Vesta £ of a year 
shorter, and Ceres and Pallas \ of a year longer than that of 
Juno, Jupiter 12 years, (11-f years,) Saturn 29| years, Uranus 
84 years, Neptune 1 64 § years. 

262. A remarkable empirical law, called Bode's Law of the 
Distances, from its discoverer, the late Professor Bode of Berlin, 
connects the distances of the planets from the sun. It is as fol- 
lows. If we take the following numbers, 0, 3, 6, 12, 24, 48, 
96, 192, and add the number 4 to each one of them, so as to ob- 
tain 4, 7, 10, 16, 28, 52, 100, 196, this series of numbers will 
express the order of distance of the planets from the sun. This 
law embodies the following curious relation between the distances 
of the orbits from one another, viz. : setting out from Venus, the 
distance between two contiguous orbits increases nearly in a dupli- 
cate ratio as we recede from the sun ; that is, the distance from 
the orbit of the earth to the orbit of Mars, is twice the distance 
from the orbit of Venus to the orbit of the earth, and one half the 
distance from the orbit of Mars to the orbits of the asteroids, &c. 
Professor Challis of Cambridge, England, has recently extended 
this principle to the distances of the satellites ; so that although 
still somewhat indefinite, it is unquestionably part of the arrange- 
ments and mechanism of the solar system.* 

Previous to the discovery of the four asteroids, to complete the 
above law a planet was wanting between Mars and Jupiter. It 
was on this account surmised by Bode, that another planet might 
exist between these two. Instead of one such planet, however, it 
was subsequently discovered that there were in fact four, revolving 
at pretty nearly the same distance from the sun, and in conformity 
with the curious law which had been detected by Bode. (Note IV.) 

263. A better idea of the dimensions of the solar system than 
is conveyed by the statement of distances above given, may be 
gained by reducing its scale sufficiently to bring it within the 
scope of familiar distances. Thus, if we suppose the earth to be 
represented by a ball only 1 inch in diameter, the distance of Mer 
cury from the sun will be represented on the same scale by 400 feet, 
the distance of Venus by 700 feet, that of the earth by 1000 feet, 
(J of a mile nearly,) that of Mars by 1500 feet, that of Juno by £ a 

* Nichol's Phenomena of the Solar System, p. 241. 



PLACE OP A PLANET IN ITS ORBIT. 



105 



mile, that of Jupiter by 1 mile, that of Saturn by 2 miles, (1 j 
miles,) and that of Uranus by 3£ miles, (3f miles.) On the same 
scale, the distance of the moon from the earth would be only 2? 
feet : that of Neptune 5| miles. 



CHAPTER VIII. 



OF THE DETERMINATION OF THE PLACE OF A PLANET, OR OF THE 
SUN, OR MOON, FOR A GIVEN TIME, BY THE ELLIPTICAL THEORY J 
AND OF THE VERIFICATION OF KEPLER's LAWS. 



PLACE OF A PLANET, OR OF THE SUN, OR MOON, IN ITS ORBIT. 

264. The angle contained between the line of apsides of a 
planet's orbit and the radius-vector, as reckoned from the peri- 
helion towards the east, is called the True Anomaly. Thus, 
let BPAP' (Fig. 53) represent Fig. 53. 

the orbit, B the perihelion, and 
P the position of the planet; 
then, BSP is its true anomaly. 
The angle contained between 
the line of apsides and the mean 
place of the planet, also reck- 
oned from the perihelion to- 
wards the east, is called the 
Mean Anomaly. Thus, let M be 
the mean place of a planet at 
the time P is its true place, and 
BSM will be its mean anomaly. 
The difference between the true anomaly BSP and the mean 
anomaly BSM, is the angular distance MSP between the true 
and mean place of the planet, or the equation of the centre, (201.) 

Describe a circle BpA on the line of apsides as a diameter ; 
through P drawpPD perpendicular to the line of apsides, and join 
p and C : the angle BCp, which the line thus determined makes 
with the line of apsides, is called the Eccentric Anomaly. 

The corresponding angles appertaining to the sun's apparent 
orbit, and to the moon's orbit, have received the same appellations. 

The interval between two consecutive returns of a body to either 
apsis of its orbit, is called the Anomalistic Revolution. The ano- 
malistic revolution of the earth, or of the sun in its apparent orbit, 
is termed, also, the Anomalistic Year. 

265. The periodic time, or the mean motion of a body, and the 
motion of the apsis of its orbit, being known, the anomalistic revo- 
lution may be easily computed. Let m = the sidereal motion of 

14 




106 DETERMINATION OF THE PLACE OF A PLANET. 

the apsis answering to the periodic time, and M = the mean daily 
motion of the planet ; then, 

M : Id. : : m : x =diff. of anomalistic rev. and periodic time. 

When the epoch of any one passage of a planet through its 
perihelion, or of the sun or moon through its perigee, has been 
found, we may, by means of the anomalistic revolution, deduce 
from it the epoch of every other passage. 

266. The length of the anomalistic year exceeds that of the 
sidereal year by 4m. 44s. 

267. From the anomalistic revolution, and the epoch of the last 
passage through the perihelion or perigee, (as the case may be,) 
we may derive the mean anomaly for any given time. Let T = 
the anomalistic revolution, t — the time that has elapsed since the 
last passage through the perihelion or perigee, and A = the mean 
anomaly : then, 

T : 360° : : t : A = 360° -^ . . . (60). 

268. The place of a body in its elliptical orbit is ascertained 
by finding its true anomaly. The problem which has for its ob- 
ject the determination of the true anomaly from the mean, was first 
resolved by Kepler, and is called Kepler's Problem. Another and 
more convenient method of obtaining the true anomaly, is to com- 
pute the equation of the centre from the mean anomaly, and add 
it to the mean anomaly, or subtract it from it, according to the po- 
sition of the body in its orbit, (201). 

HELIOCENTRIC PLACE OF A PLANET. 

269. The place of a planet in the plane of its orbit is designated 
by its orbit longitude (230) and radius-vector. To find the orbit 
longitude we have the equation V'SP = V'SB + BSP (see Fig. 
53,) or, long. = long, of perihelion + true anomaly. 

The orbit longitude may also be deduced from the mean longi- 
tude, by adding or subtracting the equation of the centre ; for, 
V'SP - V'SM + MSP, 

or, true long. = mean long. + equa. of centre : 
also, V'SP' = V'SM' - M'SP', 
or, true long. = mean long. — equa. of centre. 

The radius-vector results from the polar equation of the ellip- 
tic orbit, (235,) viz : 

V= a ^-^ . . .(61). 
1 + e cos x 

in which x denotes the true anomaly, e the eccentricity, and a the 

semi-major axis. 

270. Now to find the heliocentric longitude and latitude which 
ascertain the position of the planet with respect to the ecliptic, the 
triangle NPp (Fig. 48, p. 98) gives 

sin Pp == sin NP sin PNp ; 



GEOCENTRIC PLACE OP A PLANET. 107 

or, sin lat. = sin (oibit long. — long, of node) x sin (inclin.) . . (62); 
and 
cos PNp = tang Np cot NP, or tang Np = tang NP cos PNp, 
or, 

tang (long, — long, of node) == tang (orbit long. — long, of node) 
x cos (inclination) . . . (63). 

GEOCENTRIC PLACE OF A PLANET. 

271. From the heliocentric longitude and latitude and the radius-vector of a 
planet, to find the geocentric longitude and latitude. — Let S (Fig. 48) be the sun, 
E the earth, P the planet, jt its reduced place, and V the vernal equinox. Denote 
the heliocentric longitude VStr by L, the heliocentric latitude PSrr by I, and the 
radius-vector SP by v ; and denote the geocentric longitude by G, and the geo- 
centric latitude by A. Also let E = SE?r the elongation ; C = ES* the commu- 
tation ; A = StE the annual parallax ; and r = SE the radius-vector of the 
earth. Now, 

VE:r = SEtt -f VES, 
or, G = E + long, of sun. 

This equation will make known the geocentric longitude when the value of E 
is found. In the triangle PS* the side S* = SP cos PS7r = v cos I, and is there- 
fore known, the side ES is given by the elliptical theory, (269,) and the angle C 
may be derived from the following equation : C = VSE — VStt = long, of earth 
— long, of planet ; and to find E we have, by Trigonometry, 

ES + Stt : ES — St : : tan £ (E,rS + SE*) : tan £ (E^S — SE*,) 
or r -j- v cos I: r — v cos I : : tang £ (A -f- E) : tang £ (A — E) ; 

whence, 



1 



V cos I 



tang i (A — E) = r , V °° S ] tang £ (A + E) = 1— tang \ i4+ E). 

13 " r + » cos I " v cos I 

r 

v cos Z 

Let tang 6 = : then, 

r 

tang £ (A - E) = L^M^tang | (A + E) ; 

or, tang £ (A — E) = tang (45° — 6) tang $ ( A + E) . . . (64). 

But, A+ E = 180° — C, and E = £ (A 4- E) — J (A — E.) 

Next, to find the geocentric latitude. 

S*r tang I = P?r = Ett tang A, 

Sir tang A 

Eir ~~ tang I ' 

S;r sin E 



whence, 

but. Sjt : Err : : sin E : sin C, or 



and therefore 



Ett sin C ' 

sin E tang A 

sin C tang V 

sin E tang I 

tang A = -T--TS— • • • (65). 



sin C 

272. When a planet is in conjunction or opposition, the sines of the angles of 
elongation and commutation are each nothing. In these cases, then, the geo- 
centric latitude cannot be found by the preceding formula ; it may, however, be 
easily determined in a different manner. Suppose the planet to be in conjunction 
at P, (Fig. 49, p. 99 ;) then, 

i Ptt Pn 

tangA = _ = 1 ^-_. 



108 DETERMINATION OF THE PLACE OF A PLANET. 

But the triangle SP* gives 

Pn- = v sin I and S* = t) cos I, and ES = r ; 

, . v sin I 

hence, tang A = — ■ . . . (66).* 

r -f- v cos I s ' 

273. To find the distance of the planet from the earth, represent the distance by 
D ; then, from the triangles PttS and EPtt, (Fig. 48, p. 98,) we have 

Pir = EP sin PEjt = D sin X, 
and Ptt = SP sin PStt == » sin I ; 

whence, D = ?-^l . . . (67). 

sin X 

274. The distance of a planet being known, its horizontal parallax may be com- 
puted from the equation 

sin H = -5 . . . (68.) (Art. 113.) 



PLACES OF THE SUN AND MOON. 

275. The place of the sun, as seen from the earth, may be 
easily deduced from the heliocentric place of the earth ; for the 
longitude of the sun is equal to the heliocentric longitude of the 
earth plus 180°, (221,) and the radius-vector of the earth's orbit is 
the same as the distance of the sun from the earth. But it is more 
convenient to regard the sun as describing an orbit around the 
earth, and to compute its true anomaly, (268,) and thence tne lon- 
gitude and radius-vector by the equation 

long. = true anomaly + long, of perigee, 
and the polar equation of the orbit. 

276. The orbit longitude and the radius-vector of the moon are 
found by the same process as the longitude and radius-vector of 
the sun. The orbit longitude being known, the ecliptic longitude 
and the latitude may be determined by a process precisely similar 
to that by which the heliocentric longitude and latitude of a plane 
are found, (270.) 

VERIFICATION OF KEPLER'S LAWS. 

277. If Kepler's first two laws be true, then the geocentric 
places of the planets, computed by the process that we have 
described, (271,) which is founded upon them, ought to agree 
with the true geocentric places as obtained for the same time by 
direct observation ; or, the heliocentric places computed from the 
observed geocentric places, (242,) ought to agree with the same 
as computed by the elliptic theory, (269, 270.) Now, a great 
number of comparisons have been made between the observed and 
computed places, and in every instance a close agreement between 
the two has been found to subsist. We infer, therefore, that the 

* For inferior conjunction the sign of cos I must be changed, and for opposition 
the sign of r must be changed. 



109 

motions of the planets must be very nearly in conformity with 
these laws. 

The truth of the third law has been established by a direct 
comparison of the mean distances of the different planets with 
their periodic times. 

278. Kepler's laws have been verified for the sun and moon, in 
a similar manner. 

279. The relative distances of the sun, or moon, at different 
times, result for this purpose, from measurements of the appa- 
rent diameter, upon the principle that any two distances are in- 
versely proportional to the corresponding apparent diameters. 
Let A = semi-diameter corresponding to the mean distance, and 
6 = semi-diameter corresponding to any distance D : then 

8 : A : : 1 : D ; whence, D =-?- . . . (69) ; 

an equation which, when A has been found, will make known the 
distance corresponding to any observed semi-diameter 8, in terms 
of the mean distance as a unit. 

Now, to find A, denote the greatest and least semi-diameters 
respectively by 8', <5", and the corresponding distances by D' and 
D", and we have 

A A 

D'=-— D" = --- 
u 8 n 8'" 

and thence, 

i(D' + D") = i(^- + |r), or, i= 4 (*+£); 

2 8' 8" 
whence, A = __ . . . (70.) 

280. The distance of the sun or moon in terms of the mean 
distance as a unit, may be found in this manner ; but it may be 
had more accurately by means of a principle which has been dis- 
covered from observation, namely, that the distance is inversely 
proportional to the square root of the daily angular motion. 



CHAPTER IX. 



OF THE INEQUALITIES OF THE MOTIONS OF THE PLANETS AND OF 
THE MOON J AND OF THE CONSTRUCTION OF TABLES FOR FINDING 
THE PLACES OF THESE BODIES. 

281. It is a general law of nature, discovered by Sir Isaac 
Newton, that bodies tend, or gravitate towards each other, with 
a force directly proportional to their masses, and inversely pro- 
portional to the square of their distance. The force which causes 
one body to gravitate towards another, is supposed to arise from a 



110 



INEQUALITIES OF THE PLANETARY MOTIONS. 



mutual attraction existing between the particles of the two bodies 
and is hence called the Attraction of Gravitation. This force of 
attraction, common to all the bodies of the Solar System, is the 
general physical cause of their motions. The sun's attraction re- 
tains the planets in their orbits, and the planets by their mutual 
attractions slightly alter each other's motions. The reasoning by 
which Newton's Tlieory of Universal Gravitation is established, 
appertains to Physical Astronomy, and will be presented in another 
part of the work. 

282. If a planet were acted on by no other force than the 
attraction of the sun, it is proved that its orbit would be accu- 
rately an ellipse, and that the areas described by its radius-vector 
in equal times, would be precisely equal. But it is in reality 
attracted by the other planets, as well as the sun, and therefore 
its actual motions camiot be in strict conformity with the laws 
of Kepler. In fact, if we descend to great accuracy, the agree- 
ment between the observed and computed places noticed in Art. 
277, is found not to be exact. The deviations from the elliptic 
motion, which are produced by the attractions of the planets, 
are called Perturbations, or, in Plane Astronomy, Inequalities. 
Although, as we have just seen, the fact of the existence of ine- 
qualities in the motions of the planets is discoverable from ob- 
servation, their laws cannot be determined without the aid of 
theory. 

283. In treating of the perturbations in the motions of one 
planet, resulting from the attractions of another, the attracting 
planet is called the Disturbing Body, and the force which pro- 
duces the perturbations the Dis- 
turbing Force. To find the dis- 
turbing force, let P (Fig. 54) be 
the planet, S the sun, and M the 
disturbing body; and let PD 
represent the attraction of M for 
the planet. Decompose PD in- 
to two forces, PE and PF, one 
of which, PE, is equal and paral- 
lel to SG, the attraction of M for 
the sun ; the other, PF, will be 
known in position and intensity. 
The two forces, PE and SG, 
being equal and parallel, they 
cannot alter the relative motion 
of the sun and planet, and ac- 
cordingly may be left out of ac- 
count : there remains, therefore, 
the component PF, which will 
be wholly effectual in disturbing 

this motion. This, then, is the disturbing force. 



Fig. 54. 




PROBLEM OF THE THREE BODIES. Ill 

It happens in the case of each planet, that the distances of some 
of the other planets are so great, that their disturbing forces are 
insensible. The attractions of these bodies for the sun and planet, 
when they are exterior to the planet, are sensibly equal and paral- 
lel. Owing to the great distance of the planets from each other, 
and the smallness of their mass compared with that of the sun, the 
disturbing force is in every instance very minute in comparison 
with the sun's attraction. 

284. It is plain that the disturbing force will, in general, be 
obliquely inclined to the perpendicular to the plane of the orbit, 
PK, the tangent to the orbit, PT, and the radius-vector, PS ; and 
may, therefore, be decomposed into forces acting along these lines. 
The component along the perpendicular will alter the latitude, and 
the two others both the longitude and radius-vector ; that along 
the tangent by changing the velocity of the planet ; and that along 
the radius-vector by changing the gravity towards the sun. It ap- 
pears, therefore, that the disturbing force produces at the same 
time perturbations or inequalities of longitude, of latitude, and of 
radius-vector. 

285. Let us now consider how these inequalities may be deter- 
mined. In the first place, the inequalities produced by each 
disturbing body, may be separately investigated upon mechanical 
principles, as if the other bodies did not exist ; for the reason that 
the effect of each disturbing body is sensibly the same that it would 
be if the other bodies did not act. That this is very nearly, if not 
quite true, may be at once inferred from the minuteness of the 
whole disturbance produced by the joint action of all the disturb- 
ing forces of the system. The problem which has for its object 
the determination of the inequalities in the motions of one body, in 
its revolution around a second, produced by the attraction of a 
third, is called the Problem of the Three Bodies. If, in the case 
of any one planet, this problem be resolved for each of the other 
bodies of the system which occasion sensible perturbations, all the 
inequalities to which the motion of the planet is subject will be- 
come known. 

286. The general solution of the problem of the three bodies, 
that is, for any mass and distance of the disturbing body, or any 
intensity of the disturbing for<^, cannot be effected in the existing 
state of the mathematical sciences. But the problem has been 
resolved for the case that presents itself in nature, in which the 
disturbing force is very minute in comparison with the central 
attraction. The results obtained by the analysis, are certain an- 
alytical expressions for the perturbations in longitude, latitude, and 
radius -vector, involving variables and constants. 

287. The general expression for the whole perturbation in longitude, due to the 
action of any one disturbing body, is 

C sin (F — P) + C sin 2 (P'— P) + C" sin 3 (F — P) + &c. . . . (71), 
in which C, C, &c., are constants, P the heliocentric longitude of the body dis- 



112 INEQUALITIES OF THE PLANETARY MOTIONS. 

turbed, and P' that of the disturbing body. The number of terms is, strictly speak- 
ing, indefinite, but they form a decreasing series, so that only a small number of 
the first terms (which will be different in different cases) need to be used. 

288. The constants C, C, &c. are to be determined from observation ; they 
may, however, be determined in the case of some of the planets from theory alone. 
The process of finding them from observation is as follows : Suppose that the earth 
is the body whose perturbations are under consideration, and let D denote the per- 
turbation in longitude, produced by the joint action of all the disturbing forces. 
Then, supposing, for the sake of simplicity, that the expression for the perturbation 
due to each disturbing body consists of but two terms, we have 

D=Csin(F— P)+C'sin2 (F — P)+csin(P"— P)+c'sin2 (P" — P)+&c. (72). 

Find, by observation, the heliocentric longitude of the earth, (189, 221,) and take 
the difference between this and the longitude as computed for the same time by 
the elliptical theory, (269.) This difference will be the value of D at the time of 
the observation. P, F, P", &c. the heliocentric longitudes of the earth and of the 
disturbing bodies, and consequently P' — P, P" — P, &c, are given by the ellipti- 
cal theory. Thus, in the above equation all will be known but C, C', c, c', &c. 
By repetitions of this process as many equations may be obtained as there are con- 
stants to be determined, and from these the values of the constants may be com- 
puted. But it is usual to obtain a much greater number of equations than there 
are constants ; as, by combining them according to certain rules, much more ex- 
act values of the constants may be derived. 

289. In the expression 

C sin (F — P) + C sin 2 (F — P) + &c, 

for the perturbation in longitude, due to the action of a disturbing body, each term, 
C sin (F — P), C sin 2 (F — P), &c, is technically termed an Equation,* and is 
considered as representing a specific inequality. The angle P' — P, or 2 (P' — P), 
or other multiple of P' — P, the sine of which enters into the equation of an ine- 
quality, is called the Argument of the inequality ; and the constant is called the 
Coefficient of the inequality. As the greatest value of the sine of the argument is 
unity, the coefficient is equal to the greatest value of the inequality. 

290. The coefficient being known, the value of the inequality at any particular 
time will become known if that of the argument be found. Now, the argument 
is the difference between the longitudes of the disturbing body and disturbed body, 
or some multiple of this difference, and may be found by the elliptical theory. In 
practice, the mean longitudes may be taken, without material error, in place of the 
true, and these are easily deduced from the mean longitudes at a given epoch, by 
means of the mean motions in longitude of the two bodies. When the values of 
all the inequalities in longitude have been separately determined, by taking their 
algebraic sum we shall have the correction to be applied to the elliptic longitude in 
order to find the exact longitude. 

291. The general expression for the total perturbation of radius. vector, due to the 
action of one body, is 

Ccos(F — P) + C cos 2 (F — P)+C"cos3(F — P) + &c. . . . (73). 

As in the expression for the perturbation of longitude, each term is called an equa- 
tion, and represents a distinct inequality, the constant being the coefficient, and 
the variable angle, the cosine of which enflrs into the equation, the argument of 
the inequality. The amounts of the different inequalities, at an assumed time, 
are computed after the same manner as those of the inequalities of longitude, and 
being added together with their algebraical signs, will give the correction to be 
applied to the elliptic radius-vector. 

292. The perturbation in latitude is very minute. The inequalities of latitude, 
as of longitude and radius- vector, are represented by equations composed of a con- 
stant coefficient and the sine or cosine of a variable argument, or of the form C 
sin A or C cos A. 

• The term equation is applied in Astronomy to all quantities added to mean 
elements in order to find the true. 



PERIODIC AND SECULAR INEQUALITIES. 113 

"293. The arguments of the inequalities we have been consider- 
ing, are angles depending upon the configurations of the disturbing 
and disturbed planets with respect to each other and the sun, and 
also, in some cases, with respect to the nodes and perihelia of their 
orbits. Whenever these configurations become the same, as they 
will periodically, the arguments, and therefore the inequalities 
themselves, will have the same value. It follows, therefore, that 
the inequalities in question are periodic, 

The interval of time in which an inequality passes through all 
its gradations of positive and negative value, is called the Period 
of the inequality. It is manifestly equal to the interval of time em- 
ployed by the argument in increasing from zero to 360° ; for, m 
this interval sin A or cos A takes all its values, both positive and 
negative, and at the expiration of it recovers the same value again. 

294. It has been stated that the elements of the elliptic orbits of 
the planets are, for the most part, subject to a slow variation from 
century to century. Investigations in Physical Astronomy have 
established that the variations of the elements are due to the action 
of the disturbing forces of the planets, and that they are not pro- 
gressive, (except in the cases of the longitude of the node and the 
longitude of the perihelion,) but are really periodic inequalities 
whose periods comprise many centuries. From the great lengths 
of their periods these inequalities are termed Secular In equalities, 
in order to distinguish them from the inequalities of the elliptic mo- 
tion, denominated Periodic Inequalities, the periods of which are 
comparatively short. 

Physical Astronomy furnishes expressions called Secular Equa- 
tions, which give the value of an element at any assumed time. 

295. The inequalities of the moon's motion arise from the dis- 
turbing action of the sun. The attractions of each of the planets 
for the moon and earth are sensibly equal and parallel. The lunar 
inequalities are investigated upon the same principle as the plan- 
etary, and are represented by equations of the same general form. 
that is, consisting of a constant coefficient and the sine or cosine 
of a variable argument. They far exceed in number and magni- 
tude those of any single planet. 

296. There are three lunar inequalities of longitude which are 
prominent above the rest, and were early discovered by observa- 
tion. 

The most considerable is called the Erection, and was discover- 
ed by Ptolemy in the first century of the Christian era. It has for 
its argument double the angular distance of the moon from the 
sun minus the mean anomaly of the moon, and amounts 
greatest to 1° 20' 30". 

The second is called the Variation, and was discovered in the 
sixteenth century by Tycho Brahe. Its argument is double the 
angular distance of the moon from the sun, and its maximum value 
is 35' 42". 

15 



114 INEQUALITIES OF THE PLANETARY MOTIONS. 

The third is denominated the Annual Equation, from the cir- 
cumstance of its period being an anomalistic year. Its argument 
is the mean anomaly of the sun. When greatest it amounts to 1 1 
12". 

297. The discovery of the other lunar inequalities (with the ex- 
ception of one inequality of latitude) is due to Physical Astronomy. 

The whole number of lunar inequalities of longitude, according 
to Burckhardt, is 34, (without taking into account the equation of 
the centre and the reduction from the orbit longitude to the eclip- 
tic longitude :) and according to Damoiseau, 45. 

298. To present now at one view the entire process of finding 
the exact heliocentric place of a planet, or the geocentric place of 
the moon, at any assumed time. 

(1.) Seek the elements of the elliptic orbit from a table of ele- 
ments, such as Table II or III, allowing for the proportional part 
of the secular variation, or (more exactly) obtain them from their 
secular equations, (294.) 

(2.) Compute the longitude, latitude, and radius-vector, by the 
elliptic theory, (269, 270.) 

(3.) Compute the values of the inequalities in longitude and lati- 
tude, and of radius-vector, by means of their equations (290, 291, 
292,) and apply them individually with their proper signs, as correc- 
tions to the elliptic values of the longitude, latitude, and radius-vector. 

299. If we suppose the sun to be in motion instead of the earth, 
its inequalities will be the same as those to which the motion of 
the earth is actually subject. 

300. When the heliocentric place of a planet has been found, its 
geocentric place, if required, may be determined by the process 
explained in Art. 271. 

CONSTRUCTION OF TABLES. 

301. The determination of the place of the sun or moon, or of a 
planet, may be greatly facilitated by the use of tables. The prin- 
ciple and mode of construction of tables adapted to this purpose 
are nearly the same for each body. 

We will first explain the mode of constructing tables for facilitating the com- 
putation of the sun's longitude. We have the equation 

True long. = rr-ean long. -f- equa. of centre -\- inequalities -f- nutation. 

If, then, tables can be constructed which will furnish by inspection the mean 
longitude, the equation of the centre, the amounts of the various inequalities in 
longitude, and the nutation in longitude, at any assumed time, we may easily find 
the true longitude at the same time. 

302. (1.) For the mean longitude. — The sun's mean motion in longitude in a 
mean tropical year, is 360°. From this we may find by proportion the mean mo- 
tions in a common year of 365 days and a bissextile year of 366 days. 

With these results, and the mean longitude for the epoch of Jan. 1, 1801, (see 
Table II,) we may easily derive the mean longitude at the beginning of each of 
the years prior and subsequent to the year 1801. The second column of Table 
XVIII. contains the mean longitude of the sun at the beginning of each of the 
years inserted in the first column. The third column of this table contains the 



TABLES FOR THE SUN'S LONGITUDE. 115 

snean longitude of the perigee at the same epochs : it was constructed by means 
of the mean longitude of the perigee found for the beginning of the year 1800, 
and its mean yearly motion in longitude, which is 61".52.* 

Having the sun's mean daily motion in longitude, (192,) we obtain by propor- 
tion the motion in any proposed number of months, days, hours, minutes, or sec- 
onds. Table XIX. contains the respective amounts of the sun's motion from the 
commencement of the year to the beginning of each month ; Table XX, the sun's 
mean motion from the beginning of any month to the beginning of any day of the 
month, and his motion for hours from 1 to 24 ; and Table XXI, the same for min- 
utes and seconds from 1 to 60. With these tables the sun's mean motion in lon- 
gitude in the interval between any given time in any year and the beginning of the 
year may be had : and if this be added to the epoch for the given year, taken out 
from Table XVIII, the result will be the mean longitude at the given time. (See 
Problem IX.) 

303. Tables XIX and XX al«o contain the motions of the sun's perigee, from 
which and the epoch given by Table XVIII results the longitude of the perigee 
at any proposed time. The longitude of the perigee is given in the Solar Tables 
for the purpose of making known the mean anomaly, the mean anomaly being 
equal to the mean longitude minus the longitude of the perigee. 

304. (2.) For the equation of the centre. — To find the equation of the centre of 
an orbit we have the following equation : 

Equa. of centre = A sin 9 -f- B sin 20 -f- C sin 3d + &c. ■; 

in which A, B, C, &c, are constants that rapidly decrease in value, and which 
may be determined for any particular orbit, and 6 the mean anomaly. Now, by 
giving to the mean anomaly 6 in this equation a series of values increasing by 
small equal differences (of 1 Q , for instance,) from zero to 360°, and computing the 
corresponding values of the equation of the centre, then registering in a column 
the different values assigned to 8, and in another column to the right of this the 
computed values of the equation of the centre, we shall obtain a table which will 
give on inspection the equation of the centre corresponding to any particular mean 
anomaly. In this manner was constructed Table XXV. In this table, however, 
for the sake of compactness, the values of the equation, instead of being register- 
ed in one column, are put in as many different columns as there may be different 
numbers of signs in the value of the mean anomaly ; each column answering to 
the particular number of signs placed at the head of it. 

If the equation of the centre at an assumed time be required, find the mean 
anomaly by the tables (303,) and with the value found for it take out the equation 
of the centre from Table XXV. 

The given quantity with which a quantity is taken from a table, is called the 
Argument of that quantity. Accordingly the mean anomaly is the argument of 
the equation of the centre in Table XXV. 

305. (3.) For the inequalities. — The equations of the inequalities, as we have 
already stated, are of the form C sin A, the argument A being the difference be- 
tween the longitude of the disturbing planet and that of the earth, or some mul- 
tiple of this difference. With the equations of the inequalities a table of each 
inequafity may be constructed, upon the same principles as Table XXV. But, as 
the expression for the whole perturbation in longitude, (28?,) produced by any one 
planet, involves only two variables, the longitude of the earth and the longitude of 
the planet, it is thought to be more convenient to have a table of double entry, 
which will give the amount of the perturbation by means of the two variables as 
arguments. Such a table may be constructed, by assigning to the longitude of the 
earth and the longitude of the disturbing planet a series of values increasing by a 
common difference, and computing with each set of the values of these quanti- 
ties, the corresponding amount of the perturbation. 

In connection with the tables of the perturbations, we must have tables that 
make known the values of the arguments at any given time. Now, the mean lon- 
gitude of the sun may be foundby the solar tables (302,) and thence the mean he- 



* The quantities in Table XVIII are called Epochs. The Epoch of a quantity 
is its value at some chosen epoch. 



116 CONSTRUCTION OF ASTRONOMICAL TABLES. 

liocentric longitude of the earth by subtracting 180° ; and the mean longitude of 
the disturbing planet may be had from similar tables. The columns of Table 
XVIII, marked I, II, III, IV, V, VI, VII, contain the arguments of all the pertur- 
bations, for the beginning of each of the years registered m the first column, e*. 
pressed in thousandth parts of a circle. Tables XIX and XX contain the varia- 
tions of the arguments for months, days, and hours. Their variations for minutes 
and seconds are too small to be taken into account. With these tables, and Table 
XVIII, the values- of the arguments at any given time may be found, and by 
means of the arguments the perturbations maybe taken from Tables XXVIlI r 
XXIX, XXX, XXXI, XXXII, and XXXIII. 

306. (4.) For the nutation. — The formula for the lunar nutation in longitude, is 
17".3 sin N — 0".2 sin 2 N, in which N denotes the supplement (to 360°) of the 
longitude of the moon's ascending node. With this formula the second column of 
Table XXVII was constructed. The value of N, in thousandth parts of a circle, 
results from Tables XVIII, XIX, and XX. The solar nutation is also given by 
Table XXVII. 

307. Tables may also be constructed that will facilitate the computation of the 
radius-vector. We have 

True rad. vector := elliptic rad. vector + perturbations. 
A table of the elliptic radius-vector may be formed by means of the polar equa- 
tion of the orbit, and tables of the perturbations from their analytical expressions, 
(291.) The tables of the perturbations will have the same arguments as the tables 
of the perturbations of longitude. 

308. Lunar and planetary tables are constructed upon the same principles as the 
solar tables we have been describing, which serve to make known the orbit longi- 
tude and radius-vector. But other tables are necessary in the case of these bodies,, 
for the computation of the ecliptic longitude and the latitude. 

309. The difference between the orbit longitude and the ecliptic longitude, is 
called the Reduction to the ecliptic. A formula for the reduction has been investi- 
gated, in which the variable is the difference between the orbit longitude and the 
longitude of the node, (or, what amounts to the same, the orbit longitude plus the 
supplement of the longitude of the node to 360°.) If this formula be reduced tc 
a table, by taking the reduction from the table and adding it to the orbit longitude,, 
we shall have the ecliptic longitude. Table LIII is a table of reduction for the moon- 

310. For the latitude, we have the equation 

True lat. = lat. in orbit -|- perturbations. 
We have already seen (270) that 

sin (latin orbit )= sin (orbit long. — long, of node) sin ^inclina.) 

A table constructed from this formula will have for its argument the orbit longi- 
tude minus the longitude of the node, which is also the argument of reduction. 
(See Table LV.) 

The tables of the perturbations in latitude are constructed upon the same prin- 
ciples as the tables of the perturbations in longitude and radius-vector. 

311. A table exhibiting the longitude and latitude, right ascen- 
sion and declination, distance, parallax, semi-diameter, &c. of the 
sun or other body, at stated periods of time, as at noon of each day 
throughout the year, is called an Ephemeris of the body. An ephe- 
meris of the sun, of the moon, and of each of the planets, is pub- 
lished for each year in advance in the Eng iish Nautical Almanac,, 
and in the Connaissance des Terns. 



FORM OF THE COMETARY ORBITS. 117 

CHAPTER X. 

OF THE MOTIONS OF THE COMETS, 

312. When first seen, a comet is ordinarily at some distance 
from the sun in the heavens, and moving towards him. After 
this it continues to approach the sun for a certain time, and then 
recedes from him to a greater or less distance, and finally disap- 
pears. In many instances comets have come so near the sun, as 
to be for a time lost in his beams. It has sometimes happened 
that a comet has not made its appearance in the firmament until 
after the time of its nearest apparent approach to the sun, and 
when it is receding from him in the heavens. This was the case 
with the great comet of 1843. It was first seen, in this country, 
in open day, on the 28th of February, in the immediate vicinity 
(within 3°) of the sun; and after this moved away from him, and 
gradually diminishing in brightness, in about a month became 
invisible. 

313. Comets resemble the planets in their changes of apparent 
place among the fixed stars, but they differ from them in never 
having been observed to perform an entire circuit of the heavens. 
Their apparent motions are also more irregular than those of the 
planets, and they are confined to no particular region of the hea- 
vens, but traverse indifferently every part. 

314. Sir Isaac Newton, from observations that had been made 
upon the remarkable comet of 1680, ascertained that this comet 
described a parabolic orbit, having the sun at its focus, or an ellip- 
tic orbit of so great an eccentricity as to be undistinguishable from 
a parabola, and that its radius-vector described equal areas in equal 
times. Since then, the orbits of about 180 comets have been 
computed, and found to be, with a few exceptions, of a parabolic 
form, or sensibly so. 

315. It was demonstrated by Newton, on the theory of gravi- 
tation, that a body projected into space, may describe about the 
sun as a focus either one of the conic sections, and that the form 
of the orbit will depend upon the projectile velocity alone. With 
one particular velocity the orbit will be a parabola ; with any less 
velocity it will be an ellipse or circle ; and with any greater velo- 
city it will be an hyperbola. Now, as there is but one velocity 
from which a parabolic orbit will result, and as any comet, which 
may have originally moved in an hyperbola, must have passed 
its perihelion, and receded beyond the limits of the solar system, 
it may be inferred, with great probability, that the orbits of the 
comets whose observed courses are not distinguishable from para- 
bolic arcs, are in fact ellipses of great eccentricity. This is the 
theory of the cometary motions proposed by Newton. 



118 OF THE MOTIONS OF THE COMETS. 

The orbits of some of the comets are known from observation 
to be very eccentric ellipses. 

316. The elements of a comets orbit are the longitude of the 
ascending node, the inclination of the orbit, the longitude of the 
perihelion, the perihelion distance, and the epoch of the perihe- 
lion passage. These make known the position and dimensions of 
the orbit on the supposition that it is a parabola, and thus apper- 
tain only to the motions of the comet for the period during which 
it is visible. 

317. Assuming that the radius-vector of a comet describes 
areas proportional to the times, the elements of its orbit may be 
computed from three observed geocentric places. But the prob- 
lem is one of considerable difficulty. 

318. Astronomers do not, in general, seek to deduce from the 
observations made during one appearance of a comet its entire 
elliptic orbit. It is impossible, from such observations, to com- 
pute the major-axis of its orbit and its period with any accuracy,, 
inasmuch as in the interval during which they are made, the comet 
describes but a small portion of its entire orbit. As examples of 
the uncertainty of such determinations, four periods have been 
found by Bessel for the comet of 1807, of which the least is 1483 
years and the greatest 1952 years ; and that of the great comet 
seen in 1811 is said to be either 2301 or 3056 years. The un- 
certainty becomes much less when the period of revolution is 
short. 

The only mode of obtaining the period of a comet's revolution 
with certainty, is by directly comparing the times of its perihelion 
passages. A comet cannot be recognised at a second appearance 
by its aspect, for this is liable to great alterations. But it may be 
identified by means of the elements of its orbit, as it is extremely 
improbable that the elements of the orbits of two different comets 
will agree throughout. This method of identifying a comet on a 
second appearance may sometimes fail of application, inasmuch 
as the orbit of a comet may experience great alterations, from the 
attractions of the planets. 

319. Owing to the great lengths of the periods of most of the 
comets, and the comparatively short interval during which their 
motions have been carefully observed, there are but three comets, 
the periods and entire orbits of which have been determined. 
These are denominated Encke's Comet, Gambarfs Comet, (some- 
times called Bidet's,) and Halley's Comet. The two former have 
never been seen, except in a very few instances, without the assist- 
ance of a telescope, but the latter, when near its perihelion, is dis- 
tinctly visible to the naked eye. 

320. Encke's Comet is so called from Professor Encke, of Ber- 
lin, who first ascertained its periodical return. It accomplishes 
its revolution in the short period of 1207 days, or about 3^ years, 
and moves in an orbit inclined under a small angle (13^°) to the 



ENCKE S COMET. 



119 



plane of the ecliptic, and whose perihelion is at the distance of 
the planet Mercury, and aphelion nearly at the distance of Jupiter. 

Fig. 55. 




(See Fig. 55.) This discovery was made on the occasion of its 
fourth recorded appearance, in 1819. Since then, it has returned 
several times to its perihelion, and in every instance very nearly 
as predicted. Its last return took place in 1848 : its next will be 
in March, 1852. This comet is also called the comet of short 
period. 

321. The motions of this comet present the anomalous fact, in the solar system, 
of a period continually diminishing, and an orbit slowly contracting from some 
other cause than the disturbing actions of the other bodies of the system. Professor 
Encke finds, that after allowance has been made for all the perturbations produced 
by the planets, the actual time of each perihelion passage anticipates the time cal- 
culated from the duration of the previous revolution about 2§ hours ; and that the 
comet now arrives at its perihelion several days (about 21) sooner than it would if 
the period had remained unaltered since the comet was first seen, in 1786. This 
continual acceleration of the time of the perihelion passage cannot be attributed 
to the disturbing attraction of some unknown body, because this attraction would 
produce other effects, which have not been noticed. Encke conceives that it can 
arise from no other cause than the action of a resisting medium, or ether, in 
space. The immediate effect of the resistance of such a medium subsisting in the 
regions of space traversed by the comet, would be to diminish the velocity in 
the orbit, which it would at first seem should delay the time of the perihelion 
passage ; but the velocity being diminished, the centrifugal force is weakened, 
and consequently, the comet is drawn nearer to the sun, and moves in an orbit 
lying within the orbit due to the sun's attraction alone : its mean distance is there- 
fore diminished, and its period shortened. We have a similar phenomenon to this 



120 OF THE MOTIONS OF THE COMETS. 

in the familiar fact of the shortening of the arc of vibration, and consequent in- 
crease of the rapidity of vibration of a pendulum, under the influence of the resist- 
ance of the air. 

322. Gambart's Comet was first seen by M. Biela, at Joseph- 
stadt in Bohemia, on the 27th of February, 1826, and ten days af- 
terwards by M. Gambart, at Marseilles. The latter calculated its 
parabolic elements from his own observations, and on inspecting 
a general table of comets discovered that the same comet had pre- 
viously appeared in 1805 and 1772. Its period is about 6f years, 
(2460 days.) Its orbit is inclined under an angle of 13° to the 
plane of the ecliptic, and has its perihelion just within the orbit of 
the earth, and aphelion beyond the orbit of Jupiter, (see Fig. 55.) 
By a remarkable coincidence, the orbit of this comet very nearly 
intersects the orbit of the earth ; — so nearly that if the two bodies 
should ever chance to arrive at the point of crossing at the same 
time, the earth would encounter a portion of the filmy mass of the 
comet. It appeared, according to the prediction, in 1832; pass- 
ing through its perihelion on the 27th of November. At its next 
and last return, in 1839, it was not seen, owing to certain unfavor- 
able circumstances, (see Art. 548.) It is announced that it will 
again return to its perihelion on the 11th of February, 1846, and 
under favorable circumstances. (See Note V.) 

Gambart's comet and Encke's comet both have a direct motion, 
or in the order of the signs. 

323. Halley's Comet is so called from Sir Edmund Halley, se- 
cond Astronomer Royal of England, who ascertained its period, 
and correctly predicted its return. From a comparison of the ele- 
ments of the orbits described by the comets of 1531, 1607, and 
1682, he concluded that the same comet had made its appearance 
in these several years, and predicted that it would again return to 
its perihelion towards the end of 1758 or the beginning of 1759. 
Previous to its appearance • Clairaut, a distinguished French as- 
tronomer, undertook the arduous task of calculating its perturba- 
tions from the disturbing actions of the planets during this and the 
preceding revolution. He found that from this cause it would be 
retarded about 618 days ; 100 days from the effect of Saturn, and 
518 days from the action of Jupiter ; and predicted that it would 
reach its perihelion within a m >nth, one way or the other, of the 
middle of April, 1759. It actually passed its perihelion on the 12th 
of March, 1759. Assuming the earth's mean distance from the 
sun to be unity, the perihelion distance of this comet is 0.6, and 
aphelion distance 35.3. Accordingly it approaches the sun to with- 
in about one half the distance of the earth, and recedes from him 
to nearly twice the distance of Uranus. (See Fig. 55.) Its period 
is about 76 years, but is liable to a variation of a year or more from 
the effect of the attractions of the planets. The inclination of its 
orbit is 18°, and its motion is retrograde. The last perihelion pas- 
sage took place on the 16th of November, 1835, within a few days 



GREAT COMET OF 1843. 



121 



of the predicted time. The next will occur about the year 1911. 
It is to be expected that the perturbations will now be determined 
with such increased accuracy that the error in the prediction of 
its next perihelion passage will be less than one day. 

324. Besides the three comets whose motions have now been described, there are 
three others, the orbits and periods of which are supposed to be known, but which 
have not as yet returned to verify the predictions concerning them. These are 
Olber's Comet of 1815, the Great Comet of 1843, and Faye's Comet or the third 
comet of 1843. The first and last are telescopic comets. (See Note VL) 

Fig. 56. 




325. Olber's Comet is believed to accomplish a revolution around the sun in 75 
years ; and to be destined to return to its perihelion early in the year 1887. 

326. The astronomers of the High School Observatory in Philadelphia, and other 
astronomers in Europe, suppose that they have identified the Great Comet of 1843 
with the comets of 1668 and 1689, and predict its return about the beginning of 
the year 1865. The probable identity of this comet with that of the year 1668, 

16 



122 OF THE MOTIONS OF THE COMETS. 

seems to be generally admitted by astronomers ; but more doubt is felt with re- 
spect to the comet of 1689. Professor Peirce, of Harvard University, contend! 
that the arguments which have been offered in support of the identity of the com- 
ets of 1843 and 1689 are insufficient; and finds, after an examination of the dif 
ferent orbits which have been calculated, that the observations are, on the whole, 
best satisfied by the elliptic orbit of the French astronomers Laugier and Mauvais, 
which answers to a revolution of 175 years. 

Fig. 56 shows the parabolic path of this comet, together with various correspond- 
ing positions of the earth and comet, n is the ascending, and n' the descending 
node : the perihelion, which is within 520,000 miles of the sun's centre, is not far 
from midway between n and n'. The inclination of the orbit is 36°. The comet 
passed its perihelion on the 27th of February, at about 5 P. M., (Philadelphia 
time.) On the 28th it was seen by day at various parts of New England, the 
East and West Indies, and the south of Europe. It was then about 3° distant 
from the sun, and of a dazzling brightness. After this it showed itself with great 
distinctness early in the evening over the western horizon ; and though growing 
fainter from night to night, as it receded from the sun, continued visible to the 
naked eye until about the 3d of April. It was followed with the telescope at the 
High School Observatory until the 10th of April. 

327. Faye's Comet has a period of only about 7 years. Its perihelion is about 
60 millions of miles without the earth's orbit, and aphelion somewhat beyond the 
orbit of Jupiter. In respect to eccentricity, its orbit holds nearly a middle place 
between those of the two comets of shortest period and the most eccentric planet- 
ary orbits, (259.) The gradation is nearly as the fractions \, £, and |. 

328. Of the 180 comets whose paths have been traced, about 
an equal number have a direct and a retrograde motion. More 
than two-thirds have the perihelia of their orbits within the orbit 
of the earth. The aphelia, except in the few instances already 
cited, are beyond the orbit of Uranus. Some have come into 
close proximity to the sun. The great comet of 1680, according 
to the computation of Newton, came 166 times nearer the sun 
than the earth is. The no less remarkable comet of 1843 seems 
to have approached still nearer to him. When at its perihelion it 
was less than 100,000 miles from the sun's surface. Its velocity 
at this time was 360 miles per second, and it accomplished a semi- 
revolution (from n to n' in Fig. 56) in the remarkably short inter- 
val of 2 hours. (See Note VII.) 

There is little reason to doubt that many of the comets recede 
tens of thousands of millions of miles from the sun before they 
begin to return to him again. The periods of most of them are 
told by centuries, and of very many of them by tens of centuries. 
The planes of the orbits are inclined under every variety of angle 
to the plane of the ecliptic. 

329. The motions of the comets are liable to great derange- 
ments, from the attractions of the planets. As their orbits cross 
the orbits of the planets, they may come into proximity to these 
bodies, and be strongly attracted by them. Halley's comet has 
already (323) furnished an illustration of this general fact. The 
comet of 1770, commonly called LexelPs comet, offers a still more 
striking example of the disturbances to which the cometary motions 
are exposed. From observations made upon this comet in the * 
year 1770, Lexell made out that its period was 5| years: still, 
though a very bright comet, it has not since been seen Burck- 



GREAT NUMBER OF COMETS 123 

hardt undertook to investigate the cause of this phenomenon, and 
found that, previous to the year 1767, the comet moved in an orbit 
which answered to a period of 50 years, and never approached near 
enough to the earth and sun to become visible. Early in the year 
1767 it came so near the planet Jupiter that his attraction changed 
its orbit to one of 5^ years. It thus became visible in 1770, and 
would have again been seen on its return to the perihelion in 1776, 
had it not been so situated with regard to the earth and sun as to 
be continually hid by the sun's rays. In the year 1779 it again 
met with Jupiter, and passed so near him that his attraction was 
two hundred times greater than the attraction of the sun. The 
consequence was that its orbit was greatly enlarged, and its period 
lengthened to 20 years ; so that it no longer comes near enough to 
the earth to be visible. 

330. The number of recorded appearances of comets is about 
500. But the actual number of cometary bodies connected with 
the solar system is undoubtedly far greater than this. 

This list comprises for the great number of years which precede the time of 
the invention of the telescope, only those comets which were very conspicuous to 
the naked eye, giving, for example, only three in the thirteenth and three in the 
fourteenth century ; and since the heavens have begun to be attentively examined 
with telescopes, from two to three comets, on an average, have made their appear- 
ance every year, of which the great majority are telescopic. The periods of these, 
as well as of the others, are, in general, of" such vast length (328) that probably 
not more than half of the whole number of comets have returned twice to their 
perihelia during the last two thousand years. From these considerations it appears 
that had the heavens been attentively surveyed with the telescope during the last 
two thousand years, as many as 2500 different cometary bodies would have been 
seen. But if we reflect that there are various causes which may tend to prevent 
a comet from being seen when present in our firmament ; as unfavorable weather, 
continued proximity to the sun, too great distance from the sun and earth, (for all 
distances seem equally probable, a priori.) want of intrinsic lustre, (for there is every 
gradation of lustre from the highest to the lowest, and the fainter comets are the 
most numerous,) &c, we shall see it to be highly probable that there are, in fact, 
many thousands of these bodies. It is not difficult to perceive, as Arago has 
shown, that the paucity of observed comets with large perihelion distances, though 
apparently, is not in fact, opposed to the natural supposition that the perihelia are 
distributed uniformly throughout the region of space which surrounds the sun, even 
beyond the orbit of the most distant planet. Taking 30 as the number of comets 
that come within the orbit of Mercury, this distinguished philosopher finds that 
upon this supposition with respect to the distribution of the perihelia, the number 
of comets which come within the precincts of the solar system is no less than 
three millions and a half. 

If the hypothesis upon which this estimate is based is anywhere near the truth, 
then by far the greater number of the comets can never be seen from the earth ; 
Cor no comet has ever been visible at the distance of the orbit of Jupiter. 



124 OP THE MOTIONS OF THE SATELLITES. 

CHAPTER XI. 

OF THE MOTIONS OF THE SATELLITES. 

331. As it has already been remarked, the planets which have 
satellites are Jupiter, Saturn, and Uranus. The number of Jupi- 
ter's satellites is four ; of Saturn's, eight ; of Uranus', six. 

332. The satellites of Jupiter are perceptible with a telescope 
of very moderate power. It is found, by repeated observations, 
that they are continually changing their positions with respect to 
one another and the planet, being sometimes all to the right of the 
planet, and sometimes all to the left of it, but more frequently 
some on each side. They are distinguished from each other by 
the distance to which they recede from the planet, that which re- 
cedes to the least distance being called the First Satellite, that 
which recedes to the next greater distance the Second, and so on. 

The satellites of Jupiter were discovered by Galileo, in the 
year 1610. 

333. The satellites of Saturn and of Uranus cannot be seen 
except through excellent telescopes. They experience changes of 
apparent position, similar to those of Jupiter's satellites. 

334. The apparent motion of Jupiter's satellites alternately from 
one side to the other of the planet, leads to the supposition that 
they actually revolve around the planet. This inference is con- 
firmed by other phenomena. While a satellite is passing from the 
eastern to the western side of the planet, a small dark spot is fre- 
quently seen crossing the disc of the planet in the same direction: 
and again, while the satellite is passing from the western to the 
eastern side, it often disappears, and after remaining for a time 
invisible, reappears at another place. These phenomena are 
easily explained, if we suppose that the planet and its satellites 
are opake bodies illuminated by the sun, and that the satellites re- 
volve around the planet from west to east. On this hypothesis, 
the dark spot seen traversing the disc of the planet, is the shadow 
cast upon it by the satellite on passing between the planet and the 
sun, and the disappearance of the satellite is an eclipse, occasioned 
by its entering the shadow of the planet. 

As the transit of the shadow occurs during the passage of the 
satellite from the eastern to the western side of the planet, and the 
eclipse of the satellite during its passage from the western to the 
eastern side, the direction of the motion must be from west to east 

335 Analogous conclusions may be drawn from similar phe- 
nomena exhibited by the satellites of Saturn. The satellites of 
Uranus also revolve around their primary, but the direction of their 
motion, as referred to the ecliptic, is from east to west. 

336. Let us now examine into the principal circumstances of 



ECLIPSES OF JUPITER S SATELLITES. 



125 



Fig. 57. 



the eclipses of Jupiter's satellites, and of the transits of their shad- 
ows across the disc of the primary. Let EE'E" (Fig. 57) repre- 
sent the orbit of the earth, PP'P" the orbit of Jupiter, and ss's" 
that of one of its satel- 
lites. Suppose that E is 
the position of the earth, 
and P that of the planet, 
and conceive two lines, 
aa', bb', to be drawn tan- 
gent to the sun and plan- 
et : then, while the satel- 
lite is moving from s to s' 
it will be eclipsed, and 
while it is moving from 
f to f its shadow will 
fall upon the planet. — 
Again, if Ee, Ee' repre- 
sent two lines drawn from 
the earth tangent to the 
planet on either side, the 
satellite will, wmile mov- 
ing from g to g', traverse 
the disc of the planet, 
and w T hile moving from h 
to h', be behind the plan- 
et, and thus concealed 
from view. It will be 
seen on an inspection of 
the figure, that during 
the motion of the earth 
from E" the position of 
opposition, to E' that of conjunction, the disappearances or immer~ 
sions of the satellite will take place on the western side of the 
planet ; and that the emersions, if visible at all, can be so only 
when the earth is so far from opposition and conjunction that the 
line Es', drawn from the earth to the point of emersion, will lie to 
the west of Ee. It will also be seen, that during the passage of 
the earth from E' to E" the emersions will take place on the east- 
ern side of the planet, and that the immersions cannot be visible, 
unless the line Fs, drawn from the earth to the point of immersion, 
passes to the east of the planet. It appears from observation that 
the immersion and emersion are never both visible at the same pe- 
riod, except in the case of the third and fourth satellites. 

If the orbits of the satellites lay in the plane of Jupiter's orbit 
an eclipse of each satellite would occur every revolution, but, in 
point of fact, they are somewhat inclined to this plane, from which 
cause the fourth satellite sometimes escapes an eclipse. 

337. The periods and other particulars of the motions of the 




126 OP THE MOTIONS OF THE SATELLITES. 

satellites, result from observations upon their eclipses. The mid- 
dle point of time between the satellite entering and emerging from 
the shadow of the primary, is the time when the satellite is in the 
direction, or nearly so, of a line joining the centres of the sun and 
primary. If the latter continued stationary, then the interval be- 
tween this and the succeeding central eclipse would be the periodic 
time of the satellite. But, the primary planet moving in its orbit, 
the interval between two successive eclipses is a synodic revolu- 
tion. The synodic revolution, however, being observed, and the 
period of the primary being known, the periodic time of the satel- 
lite may be computed. 

338. The mean motions of the satellites differ but little from 
their true motions : and hence the forms of their orbits must be 
nearly circular. The orbit, however, of the third satellite of Ju- 
piter has a small eccentricity ; that of the fourth a larger. 

339. The distances of the satellites from their primary are de- 
termined from micrometrical measurements of their apparent dis- 
tances at the times of their greatest elongations. 

A comparison of the mean distances of Jupiter's satellites with 
their periodic times, proves that Kepler's third law with respect to 
the planets applies also to these bodies ; or, that the squares of 
their sidereal revolutions are as the cubes of their mean distances 
from the primary. 

The same law also has place with the satellites of Saturn and 
Uranus. 

340. The computation of the place of a satellite for a given time, 
is effected upon similar principles with that of the place of a planet, 
The mutual attractions of Jupiter's satellites occasion sensible per- 
turbations of their motions, of which account must be taken when 
it is desired to determine their places with accuracy. 

341. Laplace has shown from the theory of gravitation, that, by 
reason of the mutual attractions of the first three of Jupiter's satel- 
lites, their mean motions and mean longitudes are permanently 
connected by the following remarkable relations. 

(1.) The mean motion of the first satellite plus twice that of the 
third is equal to three times that of the second. 

(2.) The mean longitude of the first satellite plus twice that of 
the third minus three times that of the second is equal to 180°. 

342. It follows from this last relation, that the longitudes of the 
three satellites can never be the same at the same time, and conse- 
quently that they can never be all eclipsed at once. 






SOLAR TIME. 127 

CHAPTER XII. 

ON THE MEASUREMENT OF TIME 

DIFFERENT KINDS OF TIME. 

$43. In Astronomy, as we have already stated, three kinds of 
time are used — Sidereal, True or Apparent Solar, and Mean 
Solar Time ; sidereal time being measured by the diurnal mo- 
tion of the vernal equinox, true or apparent solar time by that of 
the sun, and mean solar time by that of an imaginary sun called 
the Mean sun, conceived to move uniformly in the equator with 
the real sun's mean motion in right ascension or longitude. 

344. The sidereal day and the mean solar day are each of uni- 
form duration, but the length of the true solar day is variable, a,s 
we will now proceed to show. 

The sun's daily motion in right ascension, expressed in time, is 
equal to the excess of the solar over the sidereal day. Now this 
arc, and therefore the true solar day, varies from two causes, viz : 

(1.) The inequality of the surfs daily motion in longitude. 

(2.) The obliquity of the ecliptic to the equator. 

If the ecliptic were coincident with the equator, the daily arc of 
right ascension would be > equal to the daily arc of longitude, and 
therefore would vary between the limits 57' 11" and 61' 10", 
which would answer, respectively, to the apogee and perigee. 
But, owing to the obliquity of the ecliptic, the inclination of the 
daily arc of longitude to the equator is subject to a variation ; and 
this, it is plain, (see Fig. 39,) will be attended with a variation in 
the daily arc of right ascension. The tendency of this cause is 
obviously to make the daily arc of right ascension least at the 
equinoxes, where the obliquity of the arc of longitude is greatest, 
and greatest at the solstices, where the obliquity is least. 

345. As the length of the apparent solar day is variable, it 
cannot conveniently be employed for the expression of intervals 
of time ; moreover, a clock, to keep apparent solar time, requires 
to be frequently adjusted. These inconveniences attending the 
use of apparent solar time, led astronomers to devise a new 
method of measuring time, to which they gave the name of 
mean solar time. By conceiving an imaginary sun to move uni- 
formly in the equator with the real sun's mean motion, a day was 
obtained of which the length is invariable, and equal to the mean 
length of all the apparent solar days in a tropical year ; and by 
supposing the right ascension of this fictitious sun to be, at the 
instant of the sun's arrival at the perigee of his orbit, equal to the 
sun's true longitude, and consequently at all times equal to the 
sun's mean longitude, the time deduced from its position with re- 



128 MEASUREMENT OP TIME. 

spect to the meridian, was made to correspond very nearly with 
apparent solar time. 

346. To find the excess of the mean solar day over the sidereal 
day, we have the proportion 

360° : 24 sid. hours : : 59' 8".33 : a? =3m. 56.555s. 
A mean solar day, comprising 24 mean solar hours, is, there- 
fore, 24h. 3m. 56.555s. of sidereal time. Hence, a clock regula- 
ted to sidereal time will gain 3m. 56.555s. in a mean solar day. 

347. In order to find the expression for the sidereal day in 
mean solar time, we must use the proportion 

24h. 3m. 56.555s. : £4h. : : 24h. : x =23h. 56m. 4.092s. 
The difference between this and 24 hours is 3m. 55.908s. ; and, 
therefore, a mean solar clock will lose with respect to a sidereal 
clock, or with respect to the fixed stars, 3m. 55.908s. in a sidereal 
day, and proportionally in other intervals. This is called the daily 
acceleration of the fixed stars. 

348. To express any given period of sidereal time in mean solar time, we must 

subtract for each hour — '-—-^ = 9.83s., and for minutes and seconds in the 

same proportion. And, on the other hand, to express any given period of mean 

3m. 56.55s. 
solar time in sidereal time, we must add for each hour — -^ - = 9.86s., and 

for minutes and seconds in the same proportion. 

349. It is the practice of astronomers to adjust the sidereal clock to the motions 
of the true instead of the mean equinox. The inequality of the diurnal motion of 
this point is too small to occasion any practical inconvenience. Sidereal time, as 
determined by the position of the true equinox, will not deviate from the same as 
indicated by the position of the mean equinox, more than 2.3s. in 19 years. 

350. Another species of time, called Mean Equinoctial Time, has recently been 
introduced to some extent into astronomical calculations. Mean equinoctial time 
signifies the mean time elapsed since the instant of the Mean Vernal Equinox. Its 
use is to afford a uniform date, which shall be independent of the different me- 
ridians, and of all inequalities in the sun's motion, and shall thus save the neces- 
sity, when speaking of the time of any event's happening, of mentioning at the 
same time the place where it was observed or computed. Thus, it is the same 
thing to say that a comet passed its perihelion on January 5th, 1837, at 5h. 47m. 
0.0s., mean time at Greenwich ; at 5h. 56m. 21.5s., mean time at Paris ; or at 
1836y. 289d. 6h. 16m. 40.96s., equinoctial time ; but the former dates make the 
localities of Greenwich and Paris enter as elements of the expression ; whereas 
the latter expresses the period elapsed since an epoch common to all the world, 
and identifiable independently of all localities. By this means., all ambiguities in 
the reckoning of time are supposed to be avoided.* 

CONVERSION OF ONE SPECIES OF TIME INTO ANOTHER. 

351. The difference between the apparent and mean time is 
called the Equation of Time. The equation of time, when known, 
serves for the conversion of mean time into apparent, and the 
reverse. 

352. To find the equation of time. — The hour angle of the sun 

* (Nautical Almanac for 1837, p. 515.) 



CONVERSION OF APPARENT INTO MEAN SOLAR TIME. 



129 



(p. 15, def. 16) varies at the rate 
of 360° in a solar day, or 15° per 
solar hour. If, therefore, its value 
at any moment be divided by 15, 
the quotient will be the apparent 
time at that moment. In like man- 
ner, the hour angle of the mean 
sun, divided by 15, gives the mean 
time. Now, let the circle VSD 
(Fig. 58) represent the equator, V 
the vernal equinox, M the point of 
the equator which is on the me- 
ridian, and VS the right ascension 
of the sun, and we shall have 

MS 
appar. time 




VM — VS 



15 15 

Again, if we suppose S' to be the position of the mean sun, 
(VS' being equal to the mean longitude of the sun,) we shall have 

MS' VM — VS' 

mean time = — — = — : 

15 15 



thus, equa. of time = mean time — ap. time 



VS — VS' 
15 



..(74); 






or, the equation of time is equal to the difference between the 
sun's true right ascension and mean longitude, converted into 
time. 

This rule will require some modification if very great accuracy is desired ; for, 
in seeking an expression for the mean time, the circle VSD ought properly to be 
considered as the mean equator, answering to the mean pole, (147), and the mean 
longitude of the sun is really estimated from the mean equinox V, and ought there- 
fore to be corrected by the arc VV, or the equation of the equinoxes in right as- 
cension, (147.) 

The value of the equation of time, determined from formula 
(74), is to be applied with its sign to the apparent time to obtain 
the mean, and with the opposite sign to the mean time to obtain 
the apparent. 

A formula has been investigated, and reduced to a table, which 
makes known the equation of time by means of the sun's mean 
longitude. (See Table XII.) The value of the equation of time 
at noon, on any day of the year, is also to be found in the ephem- 
eris of the sun, published in the Nautical Almanac and othei 
works. If its value for any other time than noon be desired, it 
may be obtained by simple proportion. 

353. The equation of time is zero, or mean and true time are 
the same four times in the year, viz., about the 15th of April, 
the 15th of June, the 1st of September, and the 24th of Decem- 
ber. Its greatest additive value (to apparent time) is about 14£ 
minutes, and occurs about the 11th of February; and its greatest 

17 



130 MEASUREMENT OF TIME. 

subtractive value is about 16{ minutes, and occurs about the 3d 
of November. 

354. To convert sidereal time into mean time, and vice versa. — Making use 
of Fig. 58 already employed, the arc VM, called the Right Ascension of Mid- 
Heaven, expressed in time, is the sidereal time ; VS' is the right ascension of the 
mean sun, estimated from the true equinox, or the mean longitude of the sun cor- 
rected for the equation of the equinoxes in right ascension, (352 ;) and MS' ex- 
pressed in time, is the mean time. Let the arcs VM, MS', and VS', converted 
into time, be denoted respectively by S, M, and L. Now, 

VM = MS' -f- VS' ; 
or, S = M + L. . (75); and M = S— L.. (76). 

If M + L in equation (75) exceeds 24 hours, 24 hours must be subtracted ; and 
if L exceeds S in equation (76), 24 hours must be added to S, to render the sub- 
traction possible. 

This problem may in practice be solved most easily by means of an ephemeris 
of the sun, which gives the value of S, or the sidereal time, at the instant of mean 
noon of each day, together with a table of the acceleration of sidereal on mean 
solar time, and the corresponding table of the retardation of mean on sidereal time. 

355. The conversion of apparent time into sidereal, or sidereal time into appa- 
rent, may be effected by first obtaining the mean time, and then converting this 
into sidereal or apparent time, as the case may be. 

DETERMINATION OF THE TIME AND REGULATION OF CLOCKS 
BY ASTRONOMICAL OBSERVATIONS. 

356. The regulation of a clock consists in finding its error and 
its rate. 

357. The error of a mean solar clock is most conveniently de- 
termined from observations with a transit instrument of the time, 
as given by the clock, of the meridian passage of the sun's centre. 
The time noted will be the clock time at apparent noon, and the 
exact mean time at apparent noon may be obtained by applying to 
the apparent time (24h., or Oh. 0m. 0s.) the equation of time with 
its proper sign, which may for this purpose be taken from the 
Nautical Almanac by simple inspection. A comparison of the 
clock time with the exact mean time, will give the error of the 
clock. 

358. The daily rate of a mean solar clock may be ascertained 
by finding as above the error at two successive apparent noons. 
If the two errors are the same and lie the same way, the clock goes 
accurately to mean solar time ; if they are different, their differ- 
ence or sum, according as they lie the same or opposite ways, will 
be the daily gain or loss, as the case may be. 

359. To find the error of a sidereal clock, compute the true 
right ascension of some one of the fixed stars, (see Prob. XX],) 
and note the time of its transit ; the difference between the time 
observed and the right ascension in time will be the error. The 
error of the daily rate is determined by observing two successive 
transits of the same star. The variation of the time of the second 
transit from that of the first will be the error in question. 

The error and rate may be determined more accurately from 
observations upon several stars, taking a mean of the individual 



DETERMINATION OF TIME. 



131 



results. Stars at a distance from the pole are to be selected, foi 
reasons which have been already assigned, (58). 

360. In default of a transit instrument, the time may be obtain- 
ed and time-keepers regulated by observations made out of the 
meridian. There are two methods by which this may be accom- 
plished, called, respectively, the method of Single Altitudes, and 
the method of Double Altitudes, or of Equal Altitudes. These 
we will now explain. 

(1.) To determine the time from a measured altitude of the sun, 
or of a star, its declination and also the latitude of the place being 
given. 

Let us first suppose that the altitude of the sun is taken ; cor- 



Fig. 59. 



rect the measured altitude for re- 
fraction and parallax, and also, if 
the sextant is the instrument used, 
for the semi-diameter of the sun. 
Then, if Z (Fig. 59) represents the 
zenith, P the elevated pole, and S 
the sun ; in the triangle ZPS we 
shall know ZP = co-latitude, PS N 
= co-declination, and ZS — co-alti- 
tude, from which we may compute 
the angle ZPS (= P), which is the 
angular distance of the sun from the meridian, or, if expressed in 
time, the time of the observation from apparent noon, by the fol- 
lowing equations, (App., Resolution of oblique-angled spherical 
triangles, Case 1,) 




2& = ZP+PS + ZS 



sin' 



2 _ _sin(ft — ZP) sin(A;- 
~ sin ZP sin PS" 



co-lat. + co-dec. + co-alt. 

PS) 



(78), 



or, 



sin 2 JP 



sin (k — co-lat.) sin (k — co-dec.) 



• (77); 



(79). 



sin (co-lat.) sin (co-dec.) 

The value of P being derived from these equations and convert- 
ed into time, (see Prob. Ill,) the result will be the apparent time 
at the instant of the observation, if it was made in the afternoon ; 
if not, what remains after subtracting it from 24 hours will be the 
apparent time. The apparent time being found, the mean time 
may be deduced from it by applying the equation of time. 

A more accurate result will be obtained if several altitudes be measured, the time 
of each measurement noted, and the mean of all the altitudes taken and regarded 
as corresponding to the mean of the times. The correspondence will be sufficiently 
exact if the measurements be all made within the space of 10 or 12 minutes, and 
when the sun is near the prime vertical. If an even number of altitudes be taken, 
and alternately of the upper and lower limb, the mean of the whole will give the 
altitude of the sun's centre, without it being necessary to know his apparent semi- 
diameter. In practice, the declination of the sun may be taken for the solution of 
this problem from an ephemeris of the sun. For this purpose the time of the ob» 
servation and the longitude of the place must be approximately known. 



132 



MEASUREMENT OF TIME. 



Example, On the 1st of June, 1838, at about lOh. 45m. A. Bf- 
the altitude of the sun's lower limb was measured at New York 
with a sextant, and found to be 64° 55' 5", What was the correct 
time of the observation ? 

Measured alt. of the sun's lower limb, , 64° 5^ 5" 
Sun's semi-diam., by Conn, des Terns, . 15 47 



Appar, alt. of sun's centre, 
Parallax in alt., (Table X), , 
Refraction, (Table VIII), r 

True alt, of sun's centre, 

N. York approx, time of observation, 
Diff, of long, of Paris and N, York, 

Paris approx. time of obs., . 

Sun's declin, June 1st, M, noon at Paris, 
" June 2d, 



65 



10 52 

+ 4 

—27 



65 10 29 

lOh. 45m. 
5 5 

3 50 P. M, 

22° 2' 27" 
22 10 31 



8 4 



Change of declin. in 24 hours, 

24h.:8' 4": : 3h. 50m. : I' 17", 
Declin. June 1st, M. noon at Paris, , 22° 2' 27" 



Change of declin. in 3h. 50m., 
Declin at time of obs., 

90° 0' 0" 
Lat. of N.York, 40 42 40 



1 17 



22 3 44 



Co-lat. . 
Co-dec. . 
Co-alt, , 


, 49 17 20 . 
, 67 56 16 . 
, 24 49 31 


. ar. eo. sin. 0.1203$ 
. ar, co, sin. 0,0330$ 


k . 

h — co-lat. 

ft — co-dec 


2)142 3 7 

.71 1 33 
, 21 44 13 , . 
. . 3 5 17 . . 

|P= 9 42 7.5, . 


. '.. . sin. 9.56861 
, . sin. 8.73135 




2)18.45332 

> - . » 9 22666 




P = 19 24 15 
4 






lh. 17m. 37s, 0'" 




Equa. of time, 


10 42 23 A. M. 

— 2 34 




M. time of obs. 


10 39 49 A. Mo 





THE CALENDAR. 133 

In case the altitude of a star is taken, the value of P derived from formula (79), 
when converted into time, will express the distance in time of the star from the 
meridian, and being added to the right ascension of the star, if the observation be 
made to the westward of the meridian, or subtracted from the right ascension (in- 
creased by 24h., if necessary) if the observation be made to the eastward, will give 
the sidereal time of the observation. 

(2.) To determine the time of noon from equal altitudes of the 
sun, the times of the observations being given. 

If the sun's declination did not change while he is above the hori- 
zon, he would have equal altitudes at equal times before and after 
apparent noon. Hence, if to the time of the first observation one 
half the interval of time between the two observations should be 
added, the result would be the time of noon, as shown by the clock 
or watch employed to note the times of the observations. The 
deviation from 12 o'clock would be the error of the clock with re- 
spect to apparent time. The difference between this error and the 
equation of time would be the error of the clock with respect to 
mean time. 

But, as in point of fact the sun's declination is continually chang- 
ing, equal altitudes will not have place precisely at equal times be- 
fore and after noon, and it is therefore necessary, in order to obtain 
an exact result, to apply a correction to the time thus obtained 
This correction is called the Equation of Equal Altitudes. Tables 
have been constructed by the aid of which the equation is easily 
obtained. This is at the same time a very simple and very accu- 
rate method of finding the time and the error of a clock, 

If equal altitudes of a star should be observed, it is evident that 
half the interval of time elapsed would give the time of the star 
passing the meridian, without any correction. From this the error 
of the clock (if keeping sidereal time) may be found, as explained 
in Art 359.. 

OF THE CALENDAR. 

361. The apparent motions of the sun, which bring about the 
regular succession of day and night and the vicissitude of the sea- 
sons, and the motion of the moon to and from the sun in the heav- 
ens, attended with conspicuous and regularly recurring changes in 
her disc, furnish three natural periods for the measurement of the 
lapse of time, viz. 1, the period of the apparent revolution of the 
sun with respect to the meridian, comprising the two natural pe- 
riods of day and night, which is called the solar day ; 2, the period 
of the apparent revolution of the sun with respect to the equator, 
comprehending the four seasons, which is called the tropical year ; 
3, the period of time in which the moon passes through all her 
phases and returns to the same position relative to the sun, called 
a lunar month. The day is arbitrarily divided into twenty-four 
equal parts called hours ; the hours into sixty equal parts called 
minutes ; and the minutes into sixty equal parts called seconds. 



134 MEASUREMENT OF TIME. 

The tropical year contains 365d. 5h. 48m. 48s. The lunar month 
consists of about 29| days. The week, consisting of seven days, 
has its origin in Divine appointment alone. A Calendar is a scheme 
for taking note of the lapse of time, and fixing the dates of occur- 
rences, by means of the four periods just specified, viz. the day, 
the week, the month, and the year, or periods taken as nearly equal 
to these as circumstances will admit. Different nations have, in 
general, had calendars more or less different : and the proper ad- 
justment or regulation of the calendar by astronomical observa- 
tions has in all ages and with all nations been an object of the 
highest importance. We propose, in what follows, to explain only 
the Julian and Gregorian Calendars. 

362. The Julian calendar divides the year into 12 months, con- 
taining in all 365 days. Now, it is desirable that the calendar 
should always denote the same parts of the same season by the 
same days of the same months : that, for instance, the summer and 
winter solstices, if once happening on the 21st of June and 21st 
of December, should ever after be reckoned to happen on the same 
days ; that the date of the -sun's entering the equinox, the natural 
commencement of spring, should, if once, be always on the 20th 
of March. For thus the labors of agriculture, which really depend 
on the situation of the sun in the heavens, would be simply and 
truly regulated by the calendar. 

This would happen, if the civil year of 365 days were equal to 
the astronomical ; but the latter is greater ; therefore, if the cal- 
endar should invariably distribute the year into 365 days, it would 
fall into this kind of confusion, that in process of time, and suc- 
cessively, the vernal equinox would happen on every day of the 
civil year. Let us examine this more nearly. 

Suppose the excess of the astronomical year above the civil to 
be exactly 6 hours, and on the noon of March 20th of a certain 
year, the sun to be in the equinoctial point ; then, after the lapse 
of a civil year of 365 days, the sun would be on the meridian, but 
not in the equinoctial point ; it would be to the west of that point, 
and would have to move 6 hours in order to reach it, and to com- 
plete the astronomical or tropical year. At the completions of a 
second and a third civil year, the sun would be still more and more 
remote from the equinoctial point, and would be obliged to move, 
respectively, for 12 and 18 hours before he could rejoin it and com- 
plete the astronomical year. 

At the completion of a fourth civil year the sun would be more 
distant than on the two preceding ones from the equinoctial point. 
In order to rejoin it, and to complete the astronomical year, he 
must move for 24 hours ; that is, for one whole day. In other 
words, the astronomical year would not be completed till the be- 
ginning of the next astronomical day ; till, in civil reckoning, the 
noon of March 21 st. 

At the end of four more common civil years, the sun would be 



THE CALENDAR. 135 

in the equinox on the noon of March 22d. At the end of 8 and 

64 years, on March 23d and April 6th, respectively ; at the end 
of 736 years, the sun would be in the vernal equinox on Septem- 
ber 20th ; and in a period of about 1508 years, the sun would 
have been in every sign of the zodiac on the same day of the cal- 
endar, and in the same sign on every day. 

363. If the excess of the astronomical above the civil year were 
really what we have supposed it to be, 6 hours, this confusion of 
the calendar might be most easily avoided. It would be necessa- 
ry merely to make every fourth civil year to consist of 366 days ; 
and, for that purpose, to interpose, or to intercalate, a day in a 
month previous to March. By this intercalation, what would have 
been March 21st is called March 20th, and accordingly the sun 
would be still in the equinox on the same day of the month. 

This mode of correcting the calendar was adopted by Julius 
Caesar. The fourth year into which the intercalary day is intro- 
duced was called Bissextile ; it is now frequently called the Leap 
year. The correction is called the Julian correction, and the 
length of a mean Julian year is 365d. 6h. 

By the Julian Calendar, every year that is divisible by 4 is a 
leap year, and the rest common years. 

364. The astronomical year being equal to 365d. 5h. 48m. 47.6s., 
it is less than the mean Julian by 11m. 12.4s. or 0.007782d. The 
Julian correction, therefore, itself needs correction. The calendar 
regulated by it would, in process of time, become erroneous, and 
would require reformation. 

The intercalation of the Julian correction being too great, its 
effect would be to antedate the happening of the equinox. Thus 
(to return to the old illustration) the sun, at the completion of the 
fourth civil year, now the Bissextile, would have passed the equi- 
noctial point by a time equal to four times 0.007782d. ; at the end 
of the next Bissextile, by eight times 0.007782d. ; at the end of 
130 years, by about one day. In other words, the sun would 
have been in the equinoctial point 24 hours previously, or on the 
noon of March 19th. 

In the lapse of ages this error would continue and be increased. 
Its accumulation in 1300 years would amount to 10 days, and then 
the vernal equinox would be reckoned to happen on March 10th 

365. The error into which the calendar had fallen, and would 
continue to fall, was noticed by Pope Gregory XIII. in 1582. At 
his time the length of the year was known to greater precision than 
at the time of Julius Caesar. It was supposed equal to 365d. 5h. 
49m. 16.23s. Gregory, desirous that the vernal equinox should 
be reckoned on or near March 21st, (on which day it happened in 
the year 325, when the Council of Nice was held,) ordered that 
the day succeeding the 4th of October, 1582, instead of being 
called the 5th, should be called the 15th : thus suppressing 10 
days, which, in the interval between the years 325 and 1582, 



1 36 MEASUREMENT OF TIME . 

represented nearly the accumulation of error arising from the ex- 
cessive intercalation of the Julian correction. 

This act reformed the calendar. In order to correct it in future 
ages, it was prescribed that, at certain convenient periods, the in- 
tercalary day of the Julian correction should be omitted. Thus 
the centurial years 1700, 1800, 1900, are, according to the Julian 
calendar, Bissextiles, but on these it was ordered that the interca- 
lary day should not be inserted ; inserted again in 2000, but not 
inserted in 2100, 2200, 2300 ; and so on for succeeding centuries. 
By the Gregorian calendar, then, every centurial year that is di- 
visible by 400 is a Bissextile or Leap year, and the others common 
years. For other than centurial years, the rule is the same as with 
the Julian calendar. 

366. This is a most simple mode of regulating the calendar. It 
corrects the insufficiency of the Julian correction, by omitting, in 
the space of 400 years, 3 intercalary days. And it is easy to esti- 
mate the degree of its accuracy. For the real error of the Julian 
correction is 0.007782d. in 1 year, consequently 400 X 0.007782d. 
or3.1128d. in 400 years. Consequently, 0.1128d. or 2h. 42m. 
26s. in 400 years, or 1 day in 3546 years, is the measure of the 
degree of inaccuracy in the Gregorian correction. 

367. The Gregorian calendar was adopted immediately on its 
promulgation, in all Catholic countries, but in those where the 
Protestant religion prevailed, it did not obtain a place till some 
time after. In England, " the change of style," as it was called, 
took place after the 2d of September, 1752, eleven nominal days 
being then struck out ; so that the last day of Old Style being the 
2d, the first of New Style (the next day) was called the 14th, in- 
stead of the 3d. The same legislative enactment which estab- 
lished the Gregorian calendar in England, changed the time of the 
beginning of the year from the 25th of March to the 1st of January. 
Thus the year 1752, which by the old reckoning would have com- 
menced with the 25th of March, was made to begin with the 1 st 
of January : so that the number of the year is, for dates falling 
between the 1 st of January and the 25th of March, one greater by 
the new than by the old style. In consequence of the intercalary 
day omitted in the year 1 800, there is now, for all dates, 1 2 days 
difference between the old and new style. 

Russia is at present the only Christian country in which the 
Gregorian calendar is not used. 

368. The calendar months consist, each of them, of 30 or 31 
days, except the second month, February, which, in a common 
year, contains 28 days, and in a Bissextile, 29 days ; the interca- 
lary day being added at the last of this month. 

369. To find the number of days comprised in any number of 
civil years, multiply 365 by the number of years, and add to the 
product as many days as there are Bissextile years in the period. 



PART II. 

ON THE PHENOMENA RESULTING FROM THE MOTIONS OF THE 
HEAVENLY BODIES, AND ON THEIR APPEARANCES, DIMEN- 
SIONS, AND PHYSICAL CONSTITUTION. 



CHAPTER XIII. 

OF THE SUN AND THE PHENOMENA ATTENDING ITS APPARENT 
MOTIONS. 



INEQUALITY OF DAYS * 

370. We will first give a detailed description of the sun's ap- 
parent motion with respect to the equator, the phenomenon upon 
which the inequality of days (as well as the vicissitude of the 
seasons, soon to be treated of) immediately depends. 



Fig. 60. 



Let VEAQ (Fig. 60) represent 
the equator, VTAW (inclined to 
VEAQ, under the angle TOE, 
measured by the arc TE, equal 
to 23i°,) the ecliptic, TnZ and 
Wn'Z' the two tropics, POP' the 
axis of the heavens, and PEP'Q 
the meridian and HVRA the ho- 
rizon in one of their various po- 
sitions with respect to the other 
circle s . Ab out the 2 1 st of M arch 
the sun is in the vernal equinox 
V, crossing the equator in the 
oblique direction VS, towards the 
north and east. At this time its diurnal circle is identical with the 
equator, and it crosses the meridian at the point E, south of the 
zenith a distance ZE equal to the latitude of the place. Ad- 
vancing towards the east and north, it takes up the successive 
positions S, S', S", &c., and from day to day crosses the meridian 
at r, r', &c, farther and farther to the north. Its diurnal circles 
will be, respectively, the northern parallels of declination passing 
through S, S', S", &c, and continually more and more distant 
from the equator. The distance of the sun and of its diurnal circle 
from the equator, continues to increase until about the 21st of 
June, when he reaches the summer solstice T. At this point he 




* The day, here considered, is the interval between sunrise and sunset. 

18 



138 OF THE SUN AND ITS PHENOMENA. 

moves for a short time parallel to the equator : his declination 
changes but slightly for several days, and he crosses the meridian 
from day to day at nearly the same place. It is on this account, 
viz., because the sun seems to stand still for a time wtfh respect tc 
the equator, when at the point 90° distant from the equinox, that 
this point has received the name of solstice.* The diurnal circle 
described by the sun is now identical with the tropic of Cancer, 
TnZ, which circle is so called because it passes through T the 
beginning of the sign Cancer, and when the sun reaches it, he is 
at his northern goal, and turns about and goes towards the south.t 
The sun is, also, when at the summer solstice, at its point of near- 
est approach to the zenith of every place whose latitude ZE ex- 
ceeds the obliquity of the ecliptic TE, equal to 23|- . The distance 
ZT = ZE — ET = latitude — obliquity of ecliptic. During the 
three months following the 21st of June, the sun moves over the 
arc TA, crossing the meridian from day to day at the successive 
points r", r', &c, farther and farther to the south, and arrives at 
the autumnal equinox A about the 23d of September, when its 
diurnal circle again becomes identical with the equator. It crosses 
the equator obliquely towards the east and south, and during the 
next six months has the same motion on the south of the equator, 
that it has had during the previous six months on the north of 
the equator. It employs three months in passing over the arc 
AW, during which period it crosses the meridian each day at a 
point farther to the south than on the preceding day. At the 
winter solstice, which occurs about the 22d of December, it is 
again moving paralle] to the equator, and its diurnal circle is the 
same circle as the tropic of Capricorn. In three months more it 
passes over the arc WV, crossing the meridian at the points s", s', 
&c, so that on the 21st of March it is again at the vernal equinox. 
3%1 . To explain now the phenomenon of the inequality of days 
which obtains at all places north or south of the equator. At all 
such places, the observer is in an oblique sphere ; that is, the ce- 
lestial equator and the parallels of declination are oblique to the 
horizon. This position of the sphere is represented in Fig. 11, 
p. 21, where HOR is the horizon, QOE the equator, and ncr, sct t 
Sec, parallels of declination ; WOT is the ecliptic. It is also rep- 
resented in Fig. 60, from which Fig. 1 1 differs chiefly in this, that 
the horizon, equator, ecliptic, and parallels of declination, which 
are stere ©graphically represented as ellipses in Fig. 60, are in Fig. 
1 1 orthographically projected into right lines upon the plane of the 
meridian. Since the centres of the parallels of declination are 
situated upon the axis of the heavens, which is inclined to the 
horizon, it is plain that these parallels, as it is represented in the 
Figs., and as we have before seen, (35,) will be divided into un- 
equal parts, and that the disparity between the parts will be greater 

* Fiom Sol, the sun, and sto, to stand. t From Tpenu), to turn. 



INEQUALITY OF DAYS. 139 

in proportion as the parallel is more distant from the equator; 
also, that to the north of the equator the greater parts will lie above 
the horizon, and to the south of the equator below the horizon. 
Now, the length of the day is measured by the portion of the 
parallel to the equator, described by the sun, which. lies above the 
horizon ; and it is evident, from what has just been stated, that 
(as it is shown by the Fig.) this increases continually from the 
winter solstice W to the summer solstice T, and diminishes con- 
tinually from the summer solstice T to the winter solstice W ; 
whence it appears that the day will increase in length from the 
winter to the summer solstice, and diminish in length from the 
summer to the winter solstice. 

372. As the equator is bisected by the horizon, at the equinoxes 
the day and night must be each 12 hours long. 

373. When the sun is north of the equator, the greater part of 
its diurnal circle lies above the horizon, in northern latitudes ; and, 
therefore, from the vernal to the autumnal equinox the day is, in 
the northern hemisphere, more than 12 hours in length. On the 
other hand, when the sun is south of the equator, the greater part 
of its circle lies below the horizon, and hence from the autumnal 
to the vernal equinox the day is less than 12 hours in length. 

In the latter interval the nights will obviously, at corresponding 
periods, be of the same length as the days in the former. 

374. The variation in the length of the day in the course of the 
year, will increase with the latitude of the place ; for the greater 
is the latitude, the more oblique are the circles described by the 
sun to the horizon, and the greater is the disparity between the 
parts into which they are divided by the horizon. This will be 
obvious, on referring to Fig. 11, p. 21, where HOR, H'OR', rep- 
resent the positions of the horizons of two different places with 
respect to these circles ; H'OR' being the horizon for which the 
latitude, or the altitude of the pole, is the least. 

For the same reason, the days will be the longer as we proceed 
from the equator northward, during the period that the sun is 
north of the equinoctial, and the shorter, during the period that he 
is south of this circle. 

375. At the equator the horizon bisects all the diurnal circles, 
(36,) and consequently, the day and night are there each 12 hours 
in length throughout the year. 

376. At the arctic circle the day will be 24 hours long at the 
time of the summer solstice ; for, the polar distance of the sun 
will then be 66|°, which is the same as the latitude of the arctic 
circle ; whence it follows, that the diurnal circle of the sun at this 
epoch, will correspond to the circle of perpetual apparition for the 
parallel in question. 

On the other hand, when the sun is at the winter solstice, the 
night will be 24 hours long on the arctic circle. 

377. To the north of the arctic circle, the sun will remain con- 



140 OP THE SUN AND ITS PHENOMENA. 

tinually above the horizon during the period, before and after the 
summer solstice, that his north polar distance is less than the lati- 
tude of the place, and continually below the horizon during the 
period, about the winter solstice, that his south polar distance is 
less than the latitude of the place. 

At the north pole, as the horizon is coincident with the equator, 
(37,) the sun will be above the horizon while passing from the ver- 
nal to the autumnal equinox, and below it while passing from the 
autumnal to the vernal equinox. Accordingly, at this locality there 
will be but one day and one night in the course of a year, and each 
will be of six months' duration. 

378. The circumstances of the duration of light and darkness 
are obviously the same in the southern hemisphere as in the north- 
ern, for corresponding latitudes and corresponding declinations of 
the sun. 

379. The latitude of the place and the declination of the sun 
being given, to find the times of the sun's rising and setting and 
the length of the day. 

Fig. 61. ' LetHPR(Fig. 61) be the me- 

ridian, HMR the horizon, and BsD 
the diurnal circle described by the 
sun. The hour angle EP£, or its 
measure E£, which converted into 
time expresses the interval between 
the rising or setting of the sun and 
his passage over the meridian, is 
called the Semi-diurnal Arc. Now, 

which gives 

cos E* = — sin M* ; 
and we have, by Napier's first rule, 

sin M* = cot Ms tang ts ~ tang PMH tang EB =tang PH tang EB : 
whence, cos E£ = — tang PH tang EB, 

or, cos (semi-diurnal arc) = — tang lat. x tang dec. . . (80). 

The semi-diurnal arc (in time) expresses the apparent time of 
the sun's setting; and subtracted from 12 hours, gives the appa- 
rent time of its rising. The double of it will be the length of the day. 
In resolving this problem it will, in practice, generally answer to 
make use of the declination of the sun at noon of the given day, 
which may be taken from an ephemeris. 

Exam. 1 . Let it be required to find the apparent times of the 
sun's rising and setting and the length of the day at New York at 
the summer solstice. 

Log. tang lat. (40° 42' 40") . . . 9.93474 — 
Log. tang dec. (23° 27' 40") . . 9.63749 

Log. cos (semi-diurnal arc) . . . 9.57223 — 




TWILIGHT. 141 

Semi-diurnal arc . . . . Ill 55' 40" 

Time of sun's setting .... 7h. 27m. 43s. 
Time of sun's rising . . . . 4 32 17 

Length of day 14 55 26 

Exam. 2. What are the lengths of the longest and shortest days 
at Boston; the latitude of that place being 42° 21' 15" N.? 

Ans. 15h. 6m. 28s. and 8h. 53m. 32s. 

Exam. 3. At what hours did the sun rise and set on May 1st, 
1837, at Charleston; the latitude of Charleston being 32° 47', and 
the declination of the sun being 1 5° 6' 0" N. 1 

Ans. Time of rising, 5h. 19m. 58s. Time of setting, 6h. 40m, 
2s. 

380. To find the time of the surfs apparent rising or setting; 
the latitude of the place and the declination of the sun being given. 

At the time of his apparent rising or setting, the sun as seen from 
the centre of the earth will be below the horizon a distance sS 
(Fig. 61) equal to the refraction minus the parallax. The mean 
difference of these quantities is 33' 42". Let it be denoted by R. 
Now, to find the hour angle ZPS(=P), the triangle ZPS gives, 
(see Appendix,) 
^ _ ZP + PS + ZS _ co-lat. -f co-dec. -+(90°+R) 

, . 2lTD sin (k -ZP) sin (k -PS) 
and sin 2 ^P = . ' . 1, '-, 

sin ZP sin PS 

c - 2 , p _ sin {k - co-lat.) sin (k — co-dec.) 

or, sm" \r = : — -. — : — r — : — ; -, — r . . . ( o2). 

sin (co-lat.) sm (co-dec.) 

The value of P (in time) will be the interval between apparent 
noon and the time of the apparent rising or setting, 

If the time of the rising or setting of the upper limb of the sun, 
instead of its centre, be required, we must take for R 33' 42" + 
sun's semi-diameter, or 49' 43". 

Unless very accurate results are desired, it will be sufficient to 
take the declinations of the sun at 6 o'clock in the morning and 
evening. When the greatest precision is required, the times of true 
rising and setting must be computed by equation (80), and the de- 
clinations found for these times. 

TWILIGHT. 

381. When the sun has descended below the horizon, its rays 
still continue to fall upon a certain portion of the body of air that 
lies above it, and are thence reflected down upon the earth, so as 
to occasion a certain degree of light, which gradually diminishes as 
the sun descends farther below the horizon, and the portion of the 
air posited above the horizon, that is directly illuminated, becomes 
less. The same effect, though in a reverse order, takes place in 



142 



OP THE SUN AND ITS PHENOMENA. 



the morning previous to the sun's rising. The light thus produced 
is called the Crepusculum, or Twilight. This explanation of twi- 
light will be better understood on examining Fig. 62, where AON 
represents a portion of the earth's surface, HA;R the surface of the 

Fig. 62. 




atmosphere above it, and kmS a line drawn touching the earth and 
passing through the sun. The unshaded portion, kcK, of the body 
of air which lies above the plane of the horizon HOR, is still illu- 
minated by the sun, and shines down, by reflection, upon the 
station of the observer. As the sun descends this will decrease, 
until finally when the sun is in the direction RNS' he will illumi- 
nate directly none of that part of the atmosphere which lies above 
the horizon, and twilight will be at an end. 

382. The close of the evening twilight is marked by the ap- 
pearance of faint stars over the western horizon, and the beginning 
of the morning twilight by the disappearance of faint stars situated 
in the vicinity of the eastern horizon. It has been ascertained from 
numerous observations, that, at the beginning of the morning and 
end of the evening twilight, the sun is about 18° below the horizon. 

383. At this time, then, the angle TRS' is equal to 18°. This datum will ena- 
ble us to calculate the approximate height of the atmosphere. For if the verticals 
at O, m, and N be produced to the centre of the earth, we shall have the angle 
OCN equal to TRS', or 18°, and therefore OCR equal to 9° ; and thus the height 
of the atmosphere, mR, equal to CR — Cm, equal to secant of 9° — radius. Making 
the calculation, we find the height of the atmosphere to be about 47 miles. It is 
to be understood that this is only a rough approximation. 

It will be seen, on inspecting Fig. 62, that twilight would continue longer if the 
atmosphere were higher. 

384. The latitude of the place and the surfs declination being 
given, to find the time of the beginning or end of twilight. 

The zenith distance of the sun at the beginning of morning or 
end of evening twilight, is 90° + 18° : wherefore we may solve this 
problem by means of equations (81) and (82), taking R = 18°. 

If the time of the commencement of morning twilight be sub- 
tracted from the time of sunrise, the remainder will be the dura- 
tion of twilight. 

At the latitude 49°, the sun at the time of the summer solstice 
is only 18° below the horizon, at midnight ; for the altitude of the 



TWILIGHT. 



143 



pole at a place the latitude of which is 49°, differs only IS from 
the polar distance of the sun at this epoch. This may be illustra- 
ted by Fig. 60, taking Z as the point of passage of the sun across 
the inferior meridian, PZ=67°, and PH =49°. At this latitude, 
therefore, twilight will continue all night, at the summer solstice. 
This will be true for a still stronger reason at higher latitudes. 

385. The duration of twilight varies with the latitude of the 
place and with the time of the year. At all places in the northern 
hemisphere, the summer are longer than the winter twilights ; and 
the longest twilights take place at the summer solstice ; while the 
shortest occur when the sun has a small southern declination, dif- 
ferent for each latitude.* The summer twilights increase in length 
from the equator northward. 

These facts are consequences of the different situations with respect to the hori- 
zon of the centres of the diurnal circles described by the sun in the course of the 
year, and of the different sizes of these circles. To make this evident, let us con- 
ceive a circle to be traced in the heavens parallel to the horizon, and at the dis- 
tance of 18° below it : this is called the Crepusculum Circle. The duration of 
twilight will depend upon the number of degrees in the arc of the diurnal circle of 
the sun, comprised between the horizon and the crepusculum circle, which, for the 
sake of brevity, we will call the arc of twilight : and this will vary from the two causes 
just mentioned. For, let hkr (Fig. 63) represent the equator, and h'k'r' a diurnal 



Fig. 63. 




circle described by the sun when north 
of the equator ; and let hr, st, and h V, 
s't', be the intersections of the equator 
and diurnal circle, respectively, with the 
planes of the horizon and crepusculum 
circle. When the sun is in the equator, 
the arc of twilight is hs, and when he is 
on the parallel of declination h'k'r' it is 
AV. Draw the chords hs, h's', mn, and 
the radii cs, cs', cr', en, cp. The angle 
r'h's 1 is the half of r'cs', and the angle 
pmn is the half of pen : but r'cs' is less 
than pen, and therefore r'h's 1 is less than 
pmn. Again, chs is the half of res, and 
therefore greater than pmn, the half of 
the less angle pen. Whence it appears 
that the chord h's 1 is more oblique to the 
horizon, and therefore greater than the 
chord mn, and this more oblique and greater than the chord hs. It follows, there- 
fore, that the arc h's' is greater, and contains a greater number of degrees than the 
arc mn, and that this arc is greater than hs. Thus, as the sun recedes from the 
equator towards the north, the arc of twilight, and therefore the duration of twilight, 
increases from two causes, viz : 1st. The increase in the distance of the line of in- 
tersection of the horizon with the diurnal circle from the centre of the circle ; and, 
2d. The diminution in the size of the circle. The change will manifestly be greater 
in proportion as the latitude is greater. 

* The duration of shortest twilight is given by the following formula : 

sin 9° 

sin a = — 

cos lat. 
Twice the angle a, converted into time, expresses the duration of shortest twilight 
To find the sun's declination at the time of shortest twilight, we have 

sin dec. = — tang 9° sin lat. 
(For the investigation of this and the preceding formula, see Gummere's Astrono- 
my, pages 87 and 88.) 



144 OF THE SUN AND ITS PHENOMENA. 

When the sun is south of the equator twilight will, for the same declination, 
be shorter than when he is north of the equator, because, although the diurnal cir- 
cle will be of the same size, and its intersection with the horizon at the same dis- 
tance from its centre, the intersection with the crepusculum circle will now fall 
between the intersection with the horizon and the centre, and therefore, by what 
has just been demonstrated, the arc of twilight will be shorter. 

The shortest twilight occurs when the sun is somewhat to the south of the equa- 
tor, because the arc of twilight, for a time, decreases by reason of the diminution 
of it's obliquity to the horizon more than it increases in consequence of the decrease 
in the size of the diurnal circle. That the obliquity of the arc of twilight, or rather 
of the chord of the arc, to the horizon diminishes, for a time, when the sun gets to 
the south of the equator, will appear from this, viz. that the chord is perpendicular 
to the horizon when the centre of the diurnal circle is midway between the horizon 
and the crepusculum cjrele ; which will happen when the sun is a certain dis- 
tance south of the equator, varying with the inclination of the axis of the heavens 
to the plane of the horizon, and therefore with the latitude of the place. 

The difference in the length of the summer and winter twilights, resulting from 
the causes above specified, is augmented by the inequality in the height of the at- 
mosphere. Twilight also increases in length with the obliquity of the sphere. 

386. At the poles twilight commences about a month and a half 
before the sun appears above the horizon, and lasts about a 
month and a half after he has disappeared. For, since the hori- 
zon at the poles is identical with the celestial equator, the twilight 
which precedes the long day of six months will begin when the sun 
in approaching the equator, upon the other side, attains to a decli- 
nation of 18°, and this will be about 50 days before he reaches the 
equator and rises at the pole. In like manner the evening twilight 
continues until the sun has descended 18° below the equator. 

THE SEASONS. 

387. The amount of heat received from the sun in the course 
of 24 hours, depends upon two particulars ; the time of the sun's 
continuance above the horizon, and the obliquity of his rays at 
noon. Ey reason of the obliquity of the ecliptic, both of these cir- 
cumstances vary materially in the course of the year ; whence 
arises a variation of temperature or a change of seasons. 

388. The tropics and the polar circles divide the earth into five 
parts, called Zones, throughout each of which the yearly change 
of the temperature is occasioned by a similar change in the cir- 
cumstances upon which it depends. 

The part contained between the two tropics is called the Torrid 
Zone; the two parts between the tropics and polar circles are 
called the Temperate Zones ; and the other two parts, within the 
polar circles, are called Frigid Zones. 

389. At all places in the north temperate zone the sun will al- 
ways pass the meridian to the south of the zenith ; for the latitudes 
of all such places exceed 23^°, the greatest declination of the sun. 
(See Fig. 60.) The meridian zenith distance will be greatest at 
the winter solstice, when the sun has its greatest southern decli- 
nation, and least at the summer solstice, when the sun has its 
greatest northern declination ; and it will vary continually betweeD 



THE SEASONS. 145 

the values which obtain at these epochs. The day will be longest 
at the summer solstice, and the shortest at the winter solstice, and 
will vary in length progressively from the one date to the other. 

We infer, therefore, that throughout the zone in question the 
greatest amount of heat will be received from the sun at the sum- 
mer solstice, and the least at the winter solstice ; and that the 
amount received will gradually increase, or decrease, from one of 
these epochs to the other. The solstices are not, however, the 
epochs of maximum and minimum temperature, but are found 
from observation to precede these by about a month. The reason 
of this circumstance is, that the earth continues for a month, or 
thereabouts, after the summer solstice to receive during the day 
more heat than it loses during the night, and for about the same 
length of time after the winter solstice continues to lose during the 
night more heat than it receives during the day. 

390. Within the torrid zone the length of the day varies after 
the same manner as in the temperate zone, though in a less de- 
gree ; but the motion of the sun with respect to the zenith is 
different. At all places in the torrid zone the sun passes the me- 
ridian during a certain portion of the year to the south of the zenith, 
and during the remaining portion to the north of it ; for all places 
so situated have their zeniths between the tropics in the heavens, 
and the sun moves from one tropic to the other, and back again to 
its original position, in a tropical year. Throughout the torrid 
zone, therefore, the sun will be in the zenith twice in the course of 
the year, and will be at its maximum distance from it on the one 
side and the other at the solstices. 

An inhabitant of the equator or its vicinity, will have summer 
at the two periods when the sun is in the zenith, and winter (or a 
period of minimum temperature) both at the summer and winter 
solstice. Near the tropic there will be but little variation in the 
daily amount of heat received, during the period that the sun is 
north of the zenith. 

391. At the frigid zone a new cause of a change of temperature 
exists ; the sun remains continually above the horizon for a greater 
or less number of days about the summer solstice, and continually 
below it for the same number of days about the winter solstice. 

392. The amount of the yearly variation of temperature in- 
creases with the latitude of the place ; for the greater is the lati- 
tude the greater will be the variation in the length of the day. 
Also, the mean yearly temperature is lower as we recede from the 
equator and approach the poles ; for since the sun is, in the course 
of the year, the same length of time above the horizon, at all 
places, the mean yearly temperature must depend altogether upon 
the mean obliquity of the sun's rays at noon, and this increases 
with the latitude. 

393. The yearly change in the sun's distance from the earth has 
but little effect in producing a variation of temperature upon the 

19 



146 



OF THE SUN AND ITS PHENOMENA. 



earth's surface. The change of its heating power from this cause 
amounts to no more than /y. 

394. It is important to observe, that, although in the main cli- 
mate varies with the latitude after the manner explained in the 
foregoing articles, it is still dependent more or less upon local 
circumstances, such as the vicinity of lakes, seas, and mountains, 
prevailing winds of some particular direction, &c. 

395. In the north temperate zone, Spring, Summer, Autumn, 
and Winter, the four seasons into which the year is divided, are 
considered as respectively commencing at the times of the Ver- 
nal Equinox, Summer Solstice, Autumnal Equinox, and Winter 
Solstice. 

Let V (Fig. 64) represent the vernal, and A the autumnal equi- 
nox ; S the summer, and W the winter solstice. The perigee of 

Fig. 64. 




the sun's apparent orbit is at present about 10° 15' to the east of 
the winter solstice. Let P denote its position. The lengths of 
the seasons are, agreeably to Kepler's law of areas, respectively 
proportional to the areas VES, SEA, AEW, and WEV. Thus, 
the winter is the shortest season, and the summer the longest ; 
and spring is longer than autumn. Spring and summer, taken 
together, are about 8 days longer than autumn and winter united. 

Since the perigee of the sun's orbit has a progressive motion, 
the relative lengths of the seasons must be subject to a continual 
variation. 

396. At the beginning of the year 1800, the longitude of the 
sun's perigee was 279° 30' 8".39. If from this we take 180°, the 
longitude of the autumnal equinox, the remainder, 99° 30' 8".39, 
is the distance of the perigee from the autumnal equinox at that 
epoch. The motion of the perigee in longitude is at the rate of 
61".52 per year. Dividing 99° 30' 8".39 by 61".52, the quotient 
is 5822. Hence it appears that about 5800 years anterior to the 



DIMENSIONS OF THE SUN. 147 

year 1800, the perigee coincided with the autumnal equinox, and 
the apogee with the vernal equinox. 

397. It is important to observe that the primary cause of the 
phenomenon of change of seasons, as well as of that of the ine- 
quality of days, is the inclination of the earth's axis of rotation to 
the perpendicular to the plane of its orbit, since this is the occa- 
sion of the obliquity of the ecliptic, upon which, as we have seen, 
these phenomena immediately depend. If the axis of rotation 
were perpendicular to the plane of the orbit, there would neither 
be a change of seasons nor any inequality in the length of the days 
and nights. 

APPEARANCE, DIMENSIONS, AND PHYSICAL CONSTITUTION 

OF THE SUN. 

398. The sun presents the appearance of a luminous circular 
disc. But it does not necessarily follow from this that its surface 
is really flat ; for such is the appearance of all globular bodies 
when viewed at a great distance. It is ascertained from observa- 
tions with the telescope, that the sun has a rotatory motion : this be- 
ing the fact, its surface must in reality be of a spherical form ; for 
otherwise it would not, in presenting all its sides, always appear 
under the form of a circle. 

399. The sun's real diameter is determined from his apparent 
diameter and horizontal parallax. Fig. 65. 

Let ACB (Fig. 65) represent the 
sun or other heavenly body, and 
E the place of the earth; and 
let <5 = AEB the sun's apparent 
diameter, d = 2AS his real di- 
ameter, D = ES his distance 
from the earth, and R = the radius of the earth. We have, from 
the triangle AES, 

AS =ES sin 1AEB, or, 2AS = 2ES sin |AEB j 
and thus, d = 2D sin ^5 : 

but, (equa. 7,) D = 




sinH' 

whence, d == 2R ^^ = 2R £ = 2R JL , . (83). 

sin H H 2H v J 

The mean apparent diameter of the sun is 32' 1".8, and his 

mean horizontal parallax 8".58. Accordingly we have, for the 

real diameter of the sun, 

32' 1" 8 

d = 2R Yj¥isj = 2R x 112 ( near] y-) 

Thus the diameter of the sun is about 112 times the diameter 
of the earth. The volume of the sun then exceeds that of the 
earth nearly in the proportion 112 3 to I 3 , or 1,404,928 to 1. 



148 OF THE SUN AND ITS PHENOMENA. 

From equation (83) we may derive the proportion 
d : 2R : : 8 : 2H. 

Thus, the real diameter of a heavenly body is to the diameter 
of the earth, as the apparent diameter of the body is to double its 
horizontal parallax. 

400. When the sun is viewed with' a telescope of considerable 
power, and provided with colored glasses, black spots of an irreg- 
ular form, surrounded by a dark border of a nearly uniform shade, 

Fig. 66. called a penumbra, are often seen 

^j;,;:..;,,.;;;;.,::.,, ,-'>..,,;. on its disc, (see Fig. 66.) Some- 

•I- % ■" • times several spots are included 

V ^t-i' --JlBlfr' within the same penumbra. Their 

''•^IfSlg.:--.. : --' : ^:4^flifeK&' num ber, magnitude, and positron 
;,:! ^|||t:;% '' : ' : '- : ' : ^ : t^0;P' : '' on the disc, are extremely variable. 
--$; In some years they are very fre- 

quent, and appear in large numbers ; 
■^L|i • * n others, none whatever are seen. 
%|||(pil ; In some instances more than one 
'^ hundred, of various forms and sizes, 

have been counted. They usually 
appear in clusters, composed of various numbers, from two to sixty 
or a hundred. Their absolute magnitude is often very great. 
Spots are not unfrequently seen that subtend an angle of 1' or 60". 
Now, the apparent diameter of the earth as viewed at the distance 
of the sun, is equal to double the sun's horizontal parallax, or 17" : 
the breadth of such spots must therefore exceed three times the 
diameter of the earth, or 24,000 miles. Spots two or three times 
as large as this, or about three times as great as the entire surface 
of our globe, have been seen. 

401. The form and size of the spots are subject to rapid and 
almost incessant variations. When watched from day to day, or 
even from hour to hour, they are seen to enlarge or contract, and 
at the same time to change their forms. When a spot disappears^ 
it always contracts into a point, and vanishes before the penumbra. 
Some spots disappear almost immediately after they become visi- 
ble ; others remain for weeks, or even months. 

402. Spots and streaks more luminous than the general body 
of the sun, and of a mottled appearance, are also frequently per- 
ceived upon parts of his disc, especially in the region of large 
spots, or of extensive groups of spots, or in localities where dark 
spots subsequently make their appearance. These are called Fa- 
culm. They are chiefly to be seen near the margin of the disc. 
The penumbra which surrounds each black spot is also abruptly 
terminated by a border of light more brilliant than the rest of the 
disc. 

According to Sir John Herschel, the part of the sun's disc not 
occupied by spots is far from uniformly bright. Its ground is 
finely mottled with an appearance of minute dark dots, or pores y 



AND ROTATION. 149 

which, when attentively watched, arc found to be in a constant 
state of change. 

403. When the positions of the spots on the disc are observed 
from day to day, it is perceived that they all have a common mo- 
tion in a direction from east to west. Some of the spots close up 
and vanish before they reach the western limb ; others disappear 
at the western limb, and are never afterwards seen ; a few, after 
becoming visible at the eastern limb, have been seen to pass en- 
tirely across the disc, disappear from view at the western limb, 
and re-appear again at the eastern limb. The time employed by 
a spot in traversing the sun's disc is about 14 days. About the 
same time is occupied in passing from the western to the eastern 
limb, while it is invisible. The motions of the spots are account- 
ed for, in all their circumstances, by supposing that the sun has a 
motion of rotation from west to east, around an axis nearly per- 
pendicular to the plane of the ecliptic ; and that the spots are 
portions of the solid body of the sun. The truth of this explana- 
tion of the apparent motions of the sun's spots, is confirmed by 
the changes which are observed to take place in the magnitude 
and form of the more permanent spots during their passage across 
the disc. When they first come into view at the eastern limb, 
they appear as a narrow dark streak. As they advance towards 
the middle of the disc, they gradually open out, and increase in 
magnitude ; and after they have passed the middle of the disc, 
contract by the same degrees until they are again seen as a mere 
dark line upon the western limb. 

404. A spot returns to the same position on the disc in about 
27-i- days. This is not, however, the precise period of the sun's 
rotation ; for during this interval the sun has apparently moved 
forward nearly a sign in the ecliptic ; the spot will therefore have 
accomplished that much more than a complete revolution, when it 
is again seen by an observer on the earth in the same position on 
the disc. 

405. The apparent position of a spot with respect to the sun's 
centre may be accurately determined, from day te day, by observ- 
ing, when the sun is crossing the meridian, the right ascensions 
and declinations both of the spot and centre. From three or 
more observations of this kind the period of the sun's rotation and 
the position of his equator may be ascertained. 

The time of the sun's rotation on his axis is about 25|- days ; 
the inclination of his equator to the ecliptic 7° 30' ; and the helio 
centric longitude of the ascending node of the equator 80° 7'. 

406. It is a curious fact, that the region of the sun's spots is con 
fined within about 30° of his equator. It is only occasionally that 
spots are seen in higher latitudes than this : and none are ever seen 
farther than about 60° from the equator. 

407. The only theories relative to the physical constitution of 
the sun which deserve notice, are those of Laplace and Herschel 



1 50 OF THE SUN AND ITS PHENOMENA. 

Laplace supposed that the sun was an immense globe of solid mat- 
ter in a state of ignition, and that the spots upon his disc were large 
cavities, where there was a temporary intermission in the evolution 
of luminous matter. Sir W. Herschel was of opinion that the sun 
was an opake solid body, surrounded by a transparent atmosphere 
of tens of thousands of miles in height, within which floated at a 
height of from two to three thousand miles above the solid globe a 
stratum of self-luminous clouds, which was the source of the sun's 
light and heat, and beneath this another opake and non-luminous 
stratum, which shone only with the light received from the upper 
stratum. On this hypothesis the spots are accounted for by sup- 
posing that openings occasionally take place in the strata, through 
which the dark body of the sun is seen. The penumbra Is the por- 
tion of the obscure stratum,, situated immediately around the open- 
ing made in it. This theory seems to account for all the circum- 
stances of the aspect and variation of the form and magnitude of 
the spots, which the other does not do. 

408. That the dark spots are depressions below the luminous- surface of the sun 
was first shown by Dr. Alexander Wilson, of Glasgow. He noticed that as a large 
spot, which was seen on the sun's disc in November, 17 69, came near the western limb, 
the penumbra on the side towards the centre of the disc contracted and disappeared, 
and that afterwards the luminous matter on that side seemed to encroach upon the 
central black nucleus, while in other parts the penumbra underwent but little 
change. On the reappearance of the spot at the eastern limb, be found that the 
penumbra was again wanting on the side towards the centre of the disc ; and that 
when this part made its appearance, after the spot had advanced a short distance 
upon the disc, it was much narrower than the opposite part. These various ap- 
pearances of the spot in question- are represented in Fig. 67. Dr. Wilson drew 
from these facts, the natural conclusion, that the spots were the dark body of the 

Fig. 67. 




sun seen through excavations made in the luminous matter at the surface. The 
luminous matter he conceived to have the consistence of a fog or cloud rather than 
of a liquid ; and suggested that openings might be made in it by the working of 
some sort of elastic vapor generated within the dark globe. The penumbra sur- 
rounding each Mack spot he conjectured to be the sloping sides of the opening in 
the stratum of luminous clouds. But according to this the penumbra should 3hade 
off gradually and merge into the central black spot without presenting any defi- 
nite line of demarcation ; whereas its shade is nearly uniform throughout, and it is- 
abruptly terminated, both without and within. Herschel's theory is more com- 
plete than this, and differs from it essentially in supposing the existence of an 
opake non-luminous cloudy stratum between the luminous medium and the dark 
solid globe. It was devised, after a long and diligent inspection of all the aspects 
and phenomena of the sun's spots, to account for these in all their varieties. It 
gives a satisfactory explanation of the uniformity of shade of the penumbra, which 
l)r. Wilson's theory does not do. 

409. Herschel conceives the luminous surface of the sun to be constantly in a 
state of violent agitation, and that in comparatively limited districts it is occasion- 
ally forced up into masses or waves ol hundreds of miles in height, by powerful 



PHYSICAL CONSTITUTION OF THE SUN. 



151 



upward currents, or by the exertion of some sort of explosive energy from beneath. 
The ridges of these waves constitute the faculae, which are distinctly seen only 
when near the margin of the disc, because the waves there appear in profile, and 
when near the middle of the disc are seen in front or foreshortened. This upheav- 
ing force is supposed at times to acquire such intensity as to effect an opening both 
in the lower and the upper stratum, and disclose to view the dark body of the sun. 

410. Whatever may be the true physical constitution of the sun, the changes 
which occur upon its surface take place with a rapidity which betokens the action 
of the most powerful agents, if not the existence of the most subtle and elastic me- 
dia. Some of the spots are said to have closed at the rate of nearly a mile per 
second. The slowest motion noticed is not far from a mile per minute. But these ve- 
locities of approach of the sides of a spot are vastly exceeded by the rate of motion o.. . 
the spots themselves, which has been sometimes noticed. In two well-established in. 
stances spots have been seen to break into parts, which have then rapidly receded 
from each other while the observer was viewing them through a telescope. Some 
notion of the stupendous velocity of these changes may be obtained from the con- 
sideration that the smallest area that can be distinctly discerned upon the sun, 
even through telescopes, is a circle of 465 Lilies in diameter. 

411. There has been observed, in connection with the sun, at 
certain periods of the year, a faint light that is visible before sun- 
rise and after sunset, to which has been given the name of the Zo- 
diacal Light, from the circumstance of its being mostly compre- 
hended within the zodiac. Its color is white, and its apparent fig- 
ure that of a spindle, the base of which rests on the sun, and the 
axis of which lies in the plane of the sun's equator ; such as would 
be the appearance of a body of a lenticular shape, having its centre 
coincident with the sun and its circular edge lying in the plane of 
the sun's equator. Its length varies with the season of the year 
and the state of the atmosphere ; Fig. 68. 

being sometimes more than 100°, 
and at ether times not more than 
40° or 50°. Its breadth near the 
sun varies from 8° to 30°. It is 
nowhere abruptly terminated, but 
gradually merges into the general 
light of the sky. (See Fig. 68.) 

412. No generally received ex- 
planation of this singular phenom- 
enon has yet been given. It was 
at one time supposed to be the 
atmosphere of the sun, but Laplace 
has shown that this explanation 
is at variance with the theory of 
gravitation. He found that at the 
distance of about sixteen millions 
of miles from the sun's centre 
the centrifugal force balanced 
the gravity, and that therefore 
the suits atmosphere could not extend beyond this : but this dis- 
tance is less than one half the distance of Mercury from the sun, 
whereas the substance of the zodiacal light extends b^ond the or- 
bit of Venus, and even beyond the earth's orbit. 




152 OF THE SUN AST) ITS PHENOMENA. 

Several theories have been propounded relative to the cause of the zodiacal light 
Laplace conceived it to be a ring of nebulous, that is, cloudy and self-luminous, 
matter, encircling the sun in the plane of his equator. Professor Olmsted, of 
New Haven., has suggested that it may be a large nebulous body revolving around the 
sun in a regular orbit : and the same body as that from whieh the periodical meteoric 
showers are supposed to proceed. If we were to venture another suggestion upon 
this perplexing subject, it would be, that the substance of the zodiacaflight may be 
a certain species of matter continually in the act of flowing away from the sun 
into free space : being expelled bv some repulsive force from perhaps all parts of 
its surface, but in much the greatest quantity from the region of the spots, which 
lies about the equator. Cassini, after an attentive examination of the zodiacal light 
and the sun's spots during a series of years, conceived that he had detected a con- 
nection between these two phenomena ; that the zodiacal light was fainter in propor- 
tion as the spots were fewer in number and smaller. Thus, he remarks, that after the 
year 1688, when the zodiacal light began to grow weaker, no spots appeared upon 
the sun. He thought that this phenomenon became at times entirely invisible ; and 
that this was the case in the years 1665, 1672, and 1681. From this apparent 
connection between the two phenomena he drew the natural eoneln-sinn, that the 
substance of the zodiacal light was some emanation from the Gun's spots. The 
explosive actions, which are the most probable cause of these spots, may perhaps 
furnish the luminous matter, which may afterwards be driven off to an indefinite 
distance by some repulsive action of the sun. Certainly, if there is at the sun's 
surface any matter of the same nature as that of which the tails of comets are com- 
posed, it must be expelled by the same repulsive force that drives off this species 
of matter from the heads of comets and forms their tails. (See Art. 557.) 

413. The zodiacal light is seen most distinctly in our northern 
climates in February and March after sunset., and in October and 
November before sunrise. During the month of March it may be 
seen directed towards the star Aldebaran. In December, though 
fainter, it may often be seen both in the morning and evening. 
Also towards the summer solstice it is said to be discernible, in a 
very pure state of the atmosphere, both in the morning and even- 
ing. The reason of the variations in the distinctness of the zodia- 
cal light, is found in the change of its inclination to the horizon at 
the time of sunset or sunrise, together with the variation in the du- 
ration of twilight. As its length lies in the plane of the sun's equa- 
tor, its inclination to the horizon will be different like that of this 
plane, according to the different positions of the sun m the ecliptic. 
Since the sun's equator makes but a small angle with the ecliptic, 
at sunset, the zodiacal light will be most inclined to the horizon, 
and therefore extend higher up in the heavens, towards the vernal 
equinox, when the inclination of the ecliptic to the horizon at sun- 
set is at its maximum : and, at sunrise, it will be most inclineo to 
the horizon towards the autumnal equinox, when the inclination of 
the ecliptic to the horizon at sunrise is the greatest. The zodiacal 
light is more easily and more frequently perceived in the torrid 
zone than in these latitudes, because the ecliptic and zodiac make 
there a larger angle with the horizon, and because twilight is of 
shorter duration. 



THASES OF THE MOON. 



153 



CHAPTER XIV. 



OF THE MOON AND ITS PHENOMENA. 



PHASES OF THE MOON. 

414 The most conspicuous of the phenomena exhibited by the 
moon, is the periodical change that is observed to take place in the 
form and size of its disc. The different appearances which the 
disc presents are called the Phases of the moon. 

The phenomenon in question is a simple consequence of the 
revolution of the moon around the earth. Let E (Fig. 69) rep- 
resent the position of the earth, ABC, &c, the orbit of the moon, 

Fig. 69. 




which we will suppose for the present to lie in the plane of the 
ecliptic, and ES the direction of the sun. As the distance of the 
sun from the earth is about 400 times the distance of the moon, 
lines drawn from the sun to the different parts of the moon's orbit, 
may be considered, without material error, as parallel to each 
other. If we regard the moon as an opake non-luminous body, 
of a spherical form, that hemisphere which is turned towards the 
sun will continually be illuminated by him, and the other will be 
in the dark. Now, by virtue of the moon's motion, the enlightened 
hemisphere is presented to the earth under every variety of aspect 
in the course of a synodic revolution of the moon. Thus, when 
the moon is in conjunction, as at A, this hemisphere is turned 
entirely away from the earth, and she is invisible. Soon after 
conjunction, a portion of it on the right begins to be seen, and as 
this is comprised between the right half of the circle which limits 
the vision, and the right half of the circle which separates the en- 
lightened and dark hemispheres of the moon, called the Circle of 
Illumination, it will obviously present the appearance of a crescent 
with the horns turned from the sun, as represented at B. As the 
moon advances, more and more of the enlightened half becomes 

20 



154 OF THE MOON AND ITS PHENOMENA. 

visible, and thus the crescent enlarges, and the eastern limb be- 
comes less concave. At the point C, 90° distant from the sun, 
one half of it is seen, and the disc is a semi-circle, the eastern 
limb being a right line. Beyond this point, more than half be- 
comes visible ; the nearer half of the circle of illumination falls to 
the left of the moon's centre, as seen from the earth, and thus 
becomes convex outward. This phase of the moon is repre- 
sented at D. When the moon appears under this shape, it is said 
to be Gibbous. In advancing towards opposition, the disc will 
enlarge, and the eastern limb become continually more convex ; 
and finally at opposition, where the whole illuminated face is seen 
from the earth, it will become a full circle. From opposition to 
conjunction, the nearer half of the circle of illumination will form 
the right or western limb, and this limb will pass in the inverse 
order through the same variety of forms as the eastern limb in 
the interval between conjunction and opposition. The different 
phases are delineated in the figure. 

415. The moon's orbit is, in fact, somewhat inclined to the 
plane of the ecliptic, instead of lying in it, as we have supposed ; 
but, it is plain that its inclination cannot change the order, nor the 
period of the phases, and that it can have no other effect than to 
alter somewhat the size of the disc, at particular angular distances 
from the sun. In consequence of the smallness of the inclination, 
this alteration is too slight to be noticed. 

416. When the moon is in conjunction, it is said to be New 
Moon ; and when in opposition, Full Moon. At the time be- 
tween new and full moon when the difference of the longitudes 
of the moon and sun is 90°, it is said to be the First Quarter. 
And at the corresponding time between full and new moon, it is 
said to be the Last Quarter. In both these positions the moon 
appears as a semi-circle, and is said to be dichotomized. The 
two positions of conjunction and opposition are called Syzigies ; 
and those of the first and last quarter, Quadratures. The four 
points midway between the syzigies and quadratures are called 
Octants. 

417. The interval from new moon to new moon again, is called 
a Lunar Month, and sometimes a Lunation. 

The mean daily motion of the sun in longitude is 59' 8". 33, 
and that of the moon 13° 10' 35" .03 ; wherefore the moon sepa- 
rates from the sun at the mean rate of 12° 11' 26". 70 per day; 
and hence, to find the mean length of a lunar month, we have 
the proportion 

12° 1 1' 26".70 : Id. : : 360° : x = 29d. 12h. 44m. 2.7s. 

418. To determine the time of mean new or full moon in any 
given month. 

Let the mean longitude of the sun, and also the mean longi- 
tude of the moon, at the beginning of the year, be found, and let 






TIME OF NEW OR FULL MOON. 155 

the former be subtracted from the latter, (adding 360° if neces- 
sary ;) the remainder, which call R, will be the mean distance of 
the moon to the east of the sun, at the beginning of the year. 
As the moon separates from the sun at the mean rate of 12° 11' 

26".70 per day, will express the number of days 

and fractions of a day, which at this epoch have elapsed since the 
last new moon. This interval is called the Astronomical Epact. 
If we subtract it from 29d. 12h. 44m. 2.7s. we shall have the time 
of mean new moon in January. This being known, the time of 
mean new moon in any other month of the year results very 
readily from the known length of a lunar month. 

The time of mean new moon in any month being known, the 
time of mean full moon in the same month is obtained by the ad- 
dition or subtraction, as the case may be, of half a lunar month. 

This problem is in practice most easily resolved with the aid 
of tables. (See Problem XXVII.) 

419. The time of true new moon differs from the time of mean 
new moon, for the same reasons that the true longitudes of the 
sun and moon differ from the mean. The same is true of the 
time of true full moon. For the mode of computing the time 
of true new or full moon from that of mean new or full moon, see 
Problem XXVII. 

420. The earth, as viewed from the moon, goes through the 
same phases in the course of a lunar month that the moon does 
to an inhabitant of the earth. But, at any given time, the phase 
of the earth is just the opposite to the phase of the moon. About 
the time of new moon, the earth, then near its full, reflects so 
much light to the moon as to render the obscure part visible. 
(See Fig. 69.) 

MOON'S RISING, SETTING, AND PASSAGE OVER THE MERIDIAN, 

421. To find the time of the meridian passage of the moon on 
a given day. 

Let S and M denote, respectively, the right ascension of the 
sun, and the right ascension of the moon, at noon on the given 
day, and m, s the hourly variations of the right ascension of the sun 
and moon: also let £=the required time of the meridian passage. 
At the time t the right ascensions will be, 

For the moon . . . . M + tm, 
For the sun . . . . S + ts ; 
and, as the moon is on the meridian, the difference of these arcs 
will be equal to the hour angle t ; whence, 

t = M — S + 1 (m — s) ; 
or, if all the quantities be expressed in seconds, 

* = M — S + t— i . . . (84). 
3600 V ; 



! 56 OF THE MOON AND ITS PHENOMENA. 

Thus, we find for the time of the meridian passage, 
3600(M-S) . 

f ~3600-(m-s)--- (85) - 
The quantities M, S, m, s, are, in practice, to be taken from 
ephemerides of the sun and moon. 

Example. What was the time of the passage of the moon's centre over the 
meridian of New York, on the 1st of August, 1837 ? 

When it is noon at New York, it is 4h. 56m. 4s. at Greenwich. Now, by the 
Nautical Almanac, 

Aug. 1st, at 4h. D 's R. Ascen. 8h. 58m. 36.7s. 

at 5h. « ... 9 38.3 



lh. : 56m. 4s. : : 2m. 1.6s. : lm. 53.6s. 

Aug. 1st, at 4h. D 's R. Ascen. 8h. 58m. 36.7s. 

Variation of R. Ascen. in 56m. 4s. 1 53.6 



D 's R. Ascen. at M. Noon at N. York . 9 30.3 
Aug. 1st, 0's hourly Variation of R. Ascen. . . . 9.704s 
lh. : 4h. 56m. 4s. : : 9.704s. : 47.8s. 

Aug. 1st, M. Nocn at Greenw., 0's R. Asc. Sh. 4«>ni. 31.5s. 
Variation of R. Ascen. in 4h. 56m. 4s. 47.8 



O's R. Ascen. at M. Noon at N. York . 8 46 19.3 

Aug. 1st, M. Noon at Greenw., > 's R. Asc. 8h. 50m. 27.7s. 
Aug. 2d, " '•' « 9 38 13.7 



24 ) 47 51 .0 



Aug. 1st, D 's mean hourly Varia. of R. Asc. 1 59.6 (?n) 

« ©'s " " " 9.7 (s) 



m — s = l 49.9 = 109.9s 
By Nautical Almanac, equation of time = 5m. 59s. 
lh. : 5m. 59s. : : lm. 59.6s. : 11.9s. 

D 's R. Ascen. at M. Noon at N. York . 9h. 0m. 30.3s. 
Correction for equation of time . . — 11.9 



D 's R. Ascen. at apparent Noon at N. York 9 18.4 (M) 
0's " " " " 8 46 18.3 (S) 



M — S = 14 0.1= 840.1s. 
3600 .... log. 3.55630 
M — S = 840.1 .... log. 2.92433 
3600 — (m — s) = 3490.1 . . ar. co. log. 6.45716 



Apparent time of meridian passage, 14m. 26.5s. = 866.5s. log. 2.93779 
Equa. of time at merid. passage, 5 58 



. Mean time of meridian passage, Oh. 20m. 24s. 

The Nautical Almanac gives the time of the moon's passage over the meridian 
of Greenwich for every day of the year. From this, the time of the passage across 
the meridian of any other place may easily be determined, as follows : subtract the 
tune of the meridian passage at Greenwich on the given day, from that on the 
following day, and say, as 24h. : the difference : : the longitude of the place : a 
fourth term. This fourth term, added to the time of the meridian passage at 



moon's rising and setting. 157 

Greenwich on the given day, will give the time of the meridian passage on the 
same day at the given place. 

422. Since the moon has a motion with respect to the sun, the 
time of its rising and setting must vary from day to day. When 
first seen after conjunction, it will set soon after the sun. After 
this it will set (at a mean) about 50m. later every succeeding 
night. At the first quarter, it will set about midnight ; and at full 
moon, will set about sunrise and rise about sunset. During this 
interval it will rise in the daytime, and all along from sunrise to 
sunset. From full to new moon, it will rise at night and set 
during the day ; and the time of the rising and setting will be 
about 50m. later on every succeeding night and day ; thus, at the 
last quarter it will rise about midnight and set about midday. 

423. The daily retardation of the time of the moon's rising is, 
as just stated, at a mean, about 50 minutes ; but it varies in the 
course of a revolution from about half an hour to one hour, in 
these latitudes. The retardation of the moon's rising at the time 
of full moon, varies from one full moon to another, in the course 
of the year, between the same limits. The reason of these varia- 
tions is found in the fact, that the arc of the ecliptic (12° 11') 
through which the moon moves away from the sun in a day, is 
variously inclined to the horizon, according to its situation in the 
ecliptic, and therefore employs different intervals of time in rising 
above the horizon. This fact may be very distinctly shown by 
means of a celestial globe. It will be seen that the arc in question 
will be most oblique to the horizon, and rise in the shortest time, 
in the signs Pisces and Aries. Accordingly, the full moons which 
occur in these signs will rise with the smallest retardation fiom 
day to day. These full moons occur when the sun is in the op- 
posite signs, Virgo and Libra, that is, in September and October. 
They are called, the first the Harvest Moon, and the second the 
Hunter's Moon. The time of the moon's rising at these full 
moons will, for two or three days, be only about half an hour later 
than on the preceding day. 

424. To find the time of the moon's rising or setting on any given day.- Com- 
pute the moon's semi-diurnal arc from equation (82), or (80), according as it is 
the time of the apparent rising or setting, or the time of the true rising or setting, 
that is desired. Correct it for the moon's change of right ascension in the inter- 
val between the moon's passage over the meridian and setting, by the following 
proportion, 24h. : 24 -f- m — s (421) : : semi-diurnal arc : corrected semi-diur- 
nal arc ; and add it to the time of the moon's meridian passage, found as ex- 
plained in Art. 421. The result will be the time of the moon's setting; and 
if this be subtracted from 24 hours, the remainder will be the time of the moon's 
rising. 

In consequence of the change of the moon's declination in the interval between 
i&? rising and setting, it would be more accurate to compute the semi-diurnal arc 
separately for the moon's rising. In computing the semi-diurnal arc by equation 
(80), die declination 6 hours before or after the meridian passage may be used at 
first ; and afterwards, if a more accurate result be desired, the calculation may be 
repeated with the declination found for the computed approximate time. In equa- 
tion (81 \ R = refraction — parallax = 33' 51" — 57' 1" (at a mean) = — 23' 10' 



158 OF THE MOON AND ITS PHENOMENA. 

ROTATION AND LIBRATIONS OF THE MOON. 

425. The moon presents continually nearly the same face to- 
wards the earth ; for, the same spots are always seen in nearly 
the same position upon the disc. It follows, therefore, that it 
rotates on its axis in the same direction, and with the same angu- 
lar velocity, or nearly so, that it revolves in its orbit, and thus 
completes one rotation in the same period of time in which it ac- 
complishes a revolution in its orbit. 

426. The spots on the moon's disc, although they constantly 
preserve very nearly the same situations, are not, however, strictly 
stationary. When carefully observed, they are seen alternately 
to approach and recede from the edge. Those that are very near 
the edge successively disappear and again become visible. This 
vibratory motion of the moon's spots is called Libration. 

427. There are three librations of the moon, that is, a vibratory 
motion of its spots from three distinct causes. 

(1.) The moon's motion of rotation being uniform, small portions 
on its east and west sides alternately come into sight and disap- 
pear, in consequence of its unequal motion in its orbit. The 
periodical oscillation of the spots in an easterly and westerly direc- 
tion from this cause, is called the Libration in Longitude. 

(2.) The lunar spots have also a small alternate motion from 
north to south. This is called the Libration in Latitude, and .'s 
accounted for by supposing that the moon's axis is not exactly 
perpendicular to the plane of its orbit, and that it remains contin- 
ually parallel to itself. On this supposition we ought sometimes 
to see beyond the north, pole of the moon, and sometimes beyond 
the south pole. 

(3.) Parallax is the cause of a third libration of the moon. The 
spectator upon the earth's surface being removed from its centre, 
the point towards which the moon continually presents the same 
hemisphere, he will see portions of the moon a little different 
according to its different positions above the horizon. The diur- 
nal motion of the spots resulting from the parallax, is called the 
Diurp^l or Parallactic Libration. 

42S. The exact position of the moon's equator, like that of the 
sun's, is derived from accurate observations of the situations of 
the spots upon the disc. From calculations founded upon such 
observations, it has been ascertained that the plane of the moon's 
equator is constantly inclined to the plane of the ecliptic under an 
angle of 1° 30', and intersects it in a line which is always parallel 
to the line of the nodes. It follows from the last-mentioned cir- 
cumstance, that if a plane be supposed to pass through the centre 
of the moon, parallel to the ecliptic, it will intersect the plane of 
the moon's equator and that of its orbit in the same line in which 
these planes intersect each other. The plane in question will lie 
between the plane of the equator and that of the orbit. It will 



MOON S DIMENSIONS AND PHYSICAL CONSTITUTION. 159 

make with the first an angle of 1° 30', and with the second an 
angle cf 5° 9'. 

DIMENSIONS AND PHYSICAL CONSTITUTION OF THE MOON. 

429. The phases of the moon prove it to be an opake spherical 
body. Its diameter is found by means of equation (83), viz : 

d=2R m> 

where d denotes the diameter sought, R the radius of the earth, 6 
the apparent diameter of the moon at a given distance, and H its 
horizontal parallax at the same distance. 

The greatest equatorial horizontal parallax of the moon is 61' 
24", and the corresponding apparent diameter 33' 31" : thus we 
have 

33' 31" 3 

d^2R j^r^r, = 2R jj ( vei 7 nearl y) = 2161 miles - 

The diameter of the moon being to the diameter of the earth 
as 3 to 11, the surface of the moon is to the surface of the earth 
as 3 2 to ll 2 , or as 1 to 13 ; and the volume of the moon is to the 
volume of the earth as 3 3 to ll 3 , or as 1 to 49. 

430. When the moon is viewed with a telescope, the edge of 
the disc, which borders upon the dark portion of the face, is seen 
to be very irregular and serrated, (see Fig. 70.) It is hence in 

Fiff. 70. 




f erred that the surface of the moon is diversified with mountains 
and valleys. The truth of this inference is comirmed by the fact 
that bright insulated spots are frequently seen on the dark part of 
the face near the edge of the disc, which gradually enlarge until 
they become united to it. These bright spots are doubtless the 
tops of mountains illuminated by the sun, while the surrounding 



160 OF THE MOON AND ITS PHENOMENA. 

regions that, are less elevated are involved in darkness. The disc 
is also diversified with spots of different shapes and different de- 
grees of brightness. The brighter parts are supposed to be ele- 
vated land, and the dark to be plains, and valleys, or cavities. 

431. The number of the lunar mountains is very great. Many 
of them, by their form and grouping, furnish decided indications 
of a volcanic origin. 

From measurements made with the micrometer, of the lengths 
of their shadows, or of the distance of their summits when first 
illuminated, from the adjacent boundary of the disc, the heights of 
a number of the lunar mountains have been computed".* Accord- 
ing to Herschel, the altitude of the highest is only about If Eng- 
lish miles. But Schroeter of Lilienthal, a distinguished Seleno- 
graphist, makes the elevation of some of the lunar mountains to 
exceed 5 miles: and the more recent measurements of MM. Baer 
and Madler of Berlin lead to similar results. 

432. There are no seas nor other bodies of water upon the sur- 
face of the moon. Certain dark and apparently level parts of the 
moon were for some time supposed to be extended sheets of wa- 
ter, and, under this idea, were named by Hevelius Mare Imbrium, 
Mare Crisium, &c. : but it appears that when the boundary of 
light and darkness falls upon these supposed seas, it is still more 
or less indented at some points, and salient at others, instead of 
being, as it should be, one continuous regular curve ; besides, 
when these dark spots are viewed with good telescopes, they are 
found to contain a number of cavities, whose shadows are dis- 
tinctly perceived falling within them. The spots in question are 
therefore to be regarded as extensive plains diversified by mode- 
rate elevations and depressions. The entire absence of water also 
from the farther hemisphere of the moon may be inferred from the 
fact that the moon's face is never obscured by clouds or mists. 

433. It has long been a question among Astronomers, whether the moon has an 
atmosphere. It ?s asserted, that, if it has any, it must be exceedingly rare, or 
very limited in its extent, since it does not sensibly diminish or refract the light 
of a star seen in contact with the moon's limb ; for when a star experiences an 
occultalion by reason of the interposition of the moon between it and the eye of 
the observer, it does not disappear or undergo any diminution of lustre until the 
body of the moon reaches it, and the duration of the occupation is as it is com- 
puted, without making any allowance for the refraction of a lunar atmosphere. 
But it is maintained, on the other hand, that these facts, if allowed, are not op- 
posed to the supposition of the existence of an atmosphere of a few miles only in 
height ; and that certain phenomena which have been observed afford indubitable 
evidence of the presence of a certain limited body of air upon the moon's surface. 
Thus the celebrated Schroeter, in the course of son:e delicate observations made 
upon the crescent moon, perceived a faint grayish light extending from the horns 
of the crescent a certain distance into the dark part of the moon's face. This he 
conceived to be the moon's twilight, and hence inferred the existence of a lunar 
atmosphere. From the measurements which he made of the extent of this light 
he calculated the height of that portion of the atmosphere which was capable ot 
affecting the light of a star to be about one mile. Again, in total eclipses of the 
Bun, occasioned by the interposition of the moon, the dark body of the moon has 
been ccen surrounded by a luminous ring, which was at first most distinct at the 



161 

part where the sun was last seen, and afterwards at the part where the first ray 
darted from the sun. This is supposed to have been a lunar twilight. A similar 
phenomenon was observed in the annular eclipse of 1836, just before the comple- 
tion of the ring, at the po.nt where the junction took place. 

On the whole, it seems most probable that the moon has a small atmosphere. 



434. The surface of the moon, like that of the earth, presents the two general 
varieties of level and mountainous districts ; but it differs from the earth's surface 
in having no seas, or other bodies of water, upon it, (432,) and in being more rug- 
ged and mountainous. The comparatively level regions occupy somewhat more 
than one-third of the nearer half of the moon's surface. These are, in general, 
the darker parts of the disc. The lunar plains vary in extent from 40 or 50 miles 
to 700 miles in diameter. The mountainous formations of the other parts of the 
surface offer three marked varieties, viz: 

(1.) Insulated Mountains, which rise from plains nearly level, and which may 
be supposed to present an appearance somewhat similar to Mount Etna or the 
Peak of Teneriffe. The shadows of these mountains, in certain phases of the 
moon, are as distinctly perceived as the shadow of an upright staff when placed 
opposite to the sun.* The perpendicular altitudes of some of them, as deter- 
mined from the lengths of their shadows, are between four and five miles. Insu- 
lated mountains frequently occur in the centres of circular plains. They are 
then called Central Mountains. 

(2.) Ranges of Mountains, extending in length two or three hundred miles. 
These ranges bear a distinct resemblance to our Alps, Appenines, and Andes, but 
they are much less in extent, and do not form a very prominent feature of the 
lunar surface. Some of them appear very rugged and precipitous, and the highest 
ranges are, in some places, above four miles in perpendicular altitude. In some 
instances they run nearly in a straight line from northeast to southwest, as in that 
range called the Appenines ; in other cases they assume the form of a semicircle 
or a crescentt 

(3.) Circular Formations. The general prevalence of this remarkable class of 
mountainous formations is the great characteristic feature of the topography of 
the moon's surface. It is subdivided by late selenographists into three orders, viz: 
Walled Plains, whose diameter varies from one hundred and twenty to forty or 
fifty miles ; Ring Mountains, the diameter of which descends to ten miles ; and 
Craters, which are still smaller. The term crater is sometimes extended to all 
the varieties of circular formations. They are also sometimes called Caverns, be- 
cause their enclosed plains or bottoms are sunk considerably below the general 
level of the moon's surface. 

The different orders of the circular formations differ essentially from each other 
only in size. The principal features of their constitution are, for the most part, 
the same, and they present similar varieties. Sometimes terraces are seen going 
round the ^hole ring. At other times ranges of concentric mountains encircle 
the inner toot of the wall, leaving intermediate valleys. Again, we have a [ew 
ridges of low mountains stretching through the circle contained by the wall, but 
oftener isolated conical peaks start up, and very frequently small craters having 
on an inferior scale every attribute of the large one.J The smaller craters, 
however, offer some characteristic peculiarities. Most of them are without a 
flat bottom, and have the appearance of a hollow inverted cone with the sides 
tapering towards the centre. Some have no perceptible outer edge, their margin 
being on a level with the surrounding regions : these are called Pits. 

The bounding ridge of the lunar craters or caverns is much more precipitous 
within than without ; and the internal depth of the crater is always much lower 
than the general surface of the moon. The depth varies from one-third of a mile 
to three miles and a half. 

These curious circular formations occur at almost every part of the surface, but 
are most abundant in the southwestern regions. It is the strong reflection of their 

* Dick's Celestial Scenery, p. 256. t Ibid. p. 257. 

t Nichol's Phenomena of the S >lar System, p. 1C7. 
21 



162 



ECLIPSES OF THE SUN AND MOON. 



mountainous ridges which gives to that part of the moon's surface its superior lus- 
tre. The smaller craters occupy nearly two-fifths of the moon's visible surface- 



CHAPTER XV. 

ECLIPSES OF THE SUN AND MOON. OCCULTATIONS OF THE 

TIHEb STARS. 



435. An eclipse of a heavenly body is a privation of its light 
occasioned by the interposition of some opake body between it 
and the eye, or between it and the sun. Eclipses are divided, 
with respect to the objects eclipsed, into eclipses of the sun, 

of the moon, and of the satellites, 
(334 ;) and, with respect to circum- 
stances, into total, partial, annular, 
and central. A total eclipse is one in 
which the whole disc of the lumi- 
nary is darkened ; a partial one is 
when only a part of the disc is dark- 
ened. In an annular eclipse the 
whole is darkened, except a ring or 
annul us, which appears round the 
dark part like an illuminated border; 
the definition of a central eclipse will 
be given in another place. 

ECLIPSES OF THE MOON/. 

436. An eclipse of the moon is oc- 
casioned by an interposition of the 
body of the earth directly between the 
sun and moon, and thus intercepting 
the light of the sun ; or the moon is 
eclipsed when it passes through part 
of the shadow of the earth, as pro- 
jected from the sun. Hence it is ob- 
vious that lunar eclipses can happen 
only at the time of full moon, for it 
is then only that the earth can be be- 
tween the moon and the sun. 

437. Since the sun is much larger 
than the earth, the shadow of the earth 
must have the form of a cone, the length 
of which will depend on the relative 
magnitudes of the two bodies and their 
Let the circles AGB, agb, (Fig. 71.) 




distance from/each other 



ECLIPSES OF THE MOON — EARTH'S SHADOW. 163 

be sections of the sun and earth by a plane passing through their 
centres S and E ; Act, B6, tangents to these circles on the same 
side, and Ac?, Be, tangents on different sides. The triangular space 
<iCb will be a section of the earth's shadow or Umbra, as it is 
sometimes called. The line EC is called the Axis of the Shadow. 
If we suppose the line cp to revolve about EC, and form the sur- 
face of the frustrum of a cone, of which pedq is a section, the 
space included within that surface and exterior to the umbra, is 
called the Penumbra. It is plain that points situated within the 
umbra w T ill receive no light from the sun ; and that points situated 
within the penumbra will receive light from a portion of the sun's 
disc, and from a greater portion the more distant they are from the 
umbra. 

438. To find ike length of the earth? s shadow. — Let L= the 
length of the shadow ; R=the radius of the earth ; S == sun's ap- 
parent semi-diameter, and p = sun's parallax. The right-angled 
triangle E«C (Fig. 71) gives 

EC=-^-. 
sin hLa 

Ea=R; and ECa = SEA — EAC =6—p ; whence, 

L=-^ — . . . (86<) 

sm(d— p) ' 

As the angle (<5 — p) is only about 16', it will differ but little from 
its sine, and therefore, 

L=R^— - (nearly); 

or, if 6 and p be expressed in seconds, 

T ^2062*4^8. 

L = R — t (nearly) . . . (87). 

The shadow will obviously be the shortest when the sun is the 
nearest to the earth. We then have 5 = 16' 18", andp = 9", which 
gives L = 213R. The greatest distance of the moon is a little 
less than 64 R. It appears, then, that the earth's shadow always 
extends to more than three times the distance of the moon. 

439. Let /cM/i be a circular arc, described about E the centre 
■of the earth, and with a radius equal to the distance between the 
centres ot the earth and moon at the time of opposition. The an- 
gle MEm, the apparent semi-diameter of a section of the earth's 
shadow, made at the distance of the moon's centre, is called the 
Semi-diameter of the Earths Shadow. And the angle MEA, the 
apparent semi-diameter of a section of the penumbra, \i\ the same 
distance, is called the Semi-diameter of the Penumbra. 

440. Were the plane of the moon's orbit coincider.i w r ith the 
plane of the ecliptic, there would be a lunar eclipse at every full 
moon ; out, as it is inclined tc 't, an eclipse can happen only when 



164 



ECLIPSES OF THE SUN AND MOON, 



Fig. 72. 




the full moon takes place either in one' 
of the nodes of the moon's orbit, or so 1 
near it that the moon's latitude does not 
exceed the sum of the apparent semi-di- 
ameters of the moon and of the earth's 
shadow. This will be better understood 
on referring to Fig. 72, in which N'C 
represents a portion of the ecliptic, and 
N'M a portion of the moon's orbit, N' 
the descending node, E the earth, ES r 
ES', ES" three different directions of 
the sun, s, s', s" sections of the earth's 
shadow in the three several positions 
corresponding to these directions of the 
sun, and m, m',m" the moon in opposi- 
tion. It will be seen that the moon 
^ will not pass into the earth's shadow 
* unless at the time of opposition it is 1 
nearer to the node than the point m r , where the latitude m's' is 
equal to the sum of the semi-diameters of the moon and shadow. 

441. To determine the distance from the node, beyond which 
there can be no eclipse, we must ascertain the semi-diameter of the 
earth's shadow. Let this be denoted by A, and let P = the moon's 
parallax. 

iEm = Ema - ECw (Fig. 71) ; 

but Ema = P and ECra = <$ — p (438) ; therefore, 
MEw = A=P+ j p--a . , . (8&). 

The semi-diameter of the shadow is the least when the moon rs 
in its apogee and the sun is in its perigee, or when P has its mini- 
mum, and <5 its maximum value. In these positions of the moon 
and sun, P = 53' 48", 6=16' 18", and p = 9". Substituting, we 
obtain for the least semi-diameter of the earth's shadow 37' 39"„ 
and for its least diameter 1° 15' 18". The greatest apparent diam- 
eter of the moon is 33' 31". WxHence it appears, that the diameter 
of the ear tit's shadow is always more than twice the diameter of 
the moon. 

The mean values of P and 6 are respectively 57' 1", and 16' 1"; 
which gives for the mean semi-diameter of the earth's shadow 
41' 9". 

442. If to P -fp — 8, the semi-diameter of the earth's shadow, 
we add d, the semi-diameter of the moon, the sum P + p + d — & 
will express the greatest latitude of the moon in opposition, at which 
an eclipse can happen. 

It is easy for a given value of P +p + d — S, and for a given in- 
clination of the moon's orbit, to determine within what distance from 
the node the moon must be in order that an eclipse may take place. 
By taking the least and greatest inclinations of the orbit, the great- 



LUNAR ECLIPTIC LIMITS. 165 

est and least values of P + p -\-d— 8, and also taking into view the 
inequalities in the motions of the sun and moon, it has been found, 
that when at the time of mean full moon the difference of the mean 
longitudes of the moon and node exceeds 13° 21', there cannot be 
an eclipse ; but when this difference is less than 7° 47' there must 
be one. Between 7° 47' and 13° 21' the happening of the eclipse 
is doubtful These numbers are called the Lunar Ecliptic Limits, 

To determine at what full moons in the course of any one year 
there will be an eclipse, find the time of each mean full moon, 
(418) ; and for each of the times obtained find the mean longitude 
of the sun, and also of the moon's node, and compare the differ- 
ence of these with the lunar ecliptic limits. Should, however, the 
-difference in any instance fall between the two limits, farther cal- 
culation will be necessary. 

This problem may be solved more expeditiously by means of 
tables of the sun's mean motion with respect to the moon's node. 
(See Prob. XXVIIL) 

443* The magnitude and duration of an eclipse depend upon the 
proximity of the moon to the node at the time of opposition. In 
order that the centre of the moon may be on the same right line 
with the centres of the sun and earth, or, in technical language, 
that a central eclipse may happen, the opposition must take place 
precisely in the node. A strictly central eclipse, therefore, seldom, 
if ever, occurs. As the mean semi-diameter of the earth's shadow 
is 41' 9" (441), the mean semi-diameter of the moon 15' 33", and 
the mean hourly motion of the moon with respect to the sun 30' 
29 '\ the mean duration of a central eclipse would be about 3fh. 

444. Since the moon moves from west to east, an eclipse of the 
moon must commence on the eastern limb, and end on the western. 

445. In the investigations in Arts. 438, 441, we have supposed 
the cone of the earth's shadow to be formed b)^ lines drawn from 
the edge of the sun, and touching the earth's surface. This, prob- 
ably, is not the exact case of nature ; for the duration of the eclipse, 
and thus the apparent diameter of the earth's shadow, is found by 
observation to be somewhat greater than would result from this 
supposition. This circumstance is accounted for by supposing 
those solar rays that, from their direction, would glance by and rase 
the earth's surface, to be stopped and absorbed by the lower strata 
of the atmosphere. In such a case the conical boundary of the 
earth's shadow would be formed by certain rays exterior to the 
former, and would be larger. 

The moon in approaching and receding from the earth's total 
shadow, or umbra, passes through the penumbra, and thus its light, 
instead of being extinguished and recovered suddenly, experiences 
at the beginning of the eclipse a gradual diminution, and at the end 
a gradual increase. On this account the times of the beginning 
and end of the eclipse cannot be noted with precision, and in con- 
sequence astronomers differ as to the amount of the increase in the 



166 



ECLIPSES OF THE SUN AND MOON. 



size of the earth's shadow from the cause above mentioned. It is 
the practice, however, in computing an eclipse of the moon, to in- 
crease the semi-diameter of the shadow by a ^\ part ; or, which 
amounts to the same, to add as many seconds as the semi-d*ameter 
contains minutes. 

446. It is remarked in total eclipses of the moon, that the moon 
is not wholly invisible, but appears with a dull reddish light. 

This phenomenon is doubtless another effect of the earth's at- 
mosphere, though of a totally different nature from the preceding. 
Certain of the sun's rays, instead of being stopped and absorbed,, 
are bent from their rectilinear course by the refracting power of 
the atmosphere, so as to form a cone of faint light, interior to that 
cone which has been mathematically described as the earth's shad- 
ow, which falling upon the moon renders it visible. 

447. As an eclipse of the moon is occasioned by a real loss of 
its light, it must begin and end at the same instant, and present 
precisely the same appearance, to every spectator who sees the 
moon above his horizon during the eclipse. It will be shown that 
the case is different with eclipses of the sun. 

CALCULATION OF AN ECLIPSE OF THE MOON. 

448. The apparent distance of the centre of the moon from the 
axis of the earth's shadow, and the arcs passed over by the centre 
of the moon and the axis of the shadow during an eclipse of the 
moon, being necessarily small, they may, without material error, 
be considered as right lines. We may also consider the apparent 
motion of the sun in longitude, and the motions of the moon in 
longitude and latitude, as uniform during the eclipse. These sup- 
positions being made, the calculation of the circumstances of an 
eclipse of the moon is very simple. 

Fig. 73 




Let NF (Fig. 73) be a part of the ecliptic, N the moon's as 
cending node, NL a part of the moon's orbit, C the centre of a 
section of the earth's shadow at the moon, CK perpendicular te 
NF a circle of latitude, and C the centre of the moon at the in 
stant of opposition : then CC, which is the latitude of the moon 
in opposition, is the distance of the centres of the shadow and 
moon at that time. The moon and shadow both have a motion, 
and in the same direction, as from N towards F and L. It is the 



167 

practice, however, to regard the shadow as stationary, and to attri- 
bute to the moon a motion equal to the relative motion of the moon 
and shadow. The orbit that would be described by the moon's 
centre if it had such a motion, is called the Relative Orbit of the 
moon. Inasmuch as the circumstances of the eclipse depend al- 
together upon the relative motion of the moon and shadow, this 
mode of proceeding is obviously allowable. 

As the shadow has no motion in latitude, the relative motion of 
the moon and shadow in latitude will be equal to the moon's ac- 
tual motion in latitude : and since the centre of the earth's shadow 
moves in the plane of the ecliptic at the same rate as the sun, the 
relative motion of the moon and shadow in longitude will be equal 
to the difference between the motions of the sun and moon in lon- 
gitude. We obtain, therefore, the relative position of the centres 
of the moon and shadow at any interval t, following opposition, by 
laying off' Cm equal to the difference of the motions of the sun and 
moon in longitude in this interval, through m drawing mM per- 
pendicular to NF, and cutting off mM equal to the latitude at op- 
position plus the motion in latitude in the interval t : M will be the 
position of the moon's centre in the relative orbit, the centre of the 
shadow being supposed to be stationary at C. As the motion of 
the sun in longitude, and of the moon in longitude and latitude, is 
considered uniform, the ratio of Cm' (= Cm, the difference be- 
tween the motions of the sun and moon in longitude) to Mm 1 the 
moon's motion in latitude, is the same, whatever may be the length 
of the interval considered. It follows, therefore, that the relative 
orbit of the moon N'C'M is a right line. 

449. The relative orbit passes through C, the place of the moon's centre at op- 
position : its position will therefore be known, if its inclination to the ecliptic be 
found. Now we have 

Mm' moon's motion in latitude 

tan inclma, = -^—, = - — : — - .— 

L, m moon s mot. in long. — sun's mot. in long. 

450. The following data arc requisite in the calculation of the circumstances of 
a lunar eclipse : 

T = time of opposition. 

M = moon's hourly motion in longitude. 

n = moon's hourly motion in latitude. 

m = sun's hourly motion in longitude. 

X = moon's latitude at opposition. 

d = moon's semi-diameter. 

& = sun's semi-diameter. 

P = moon's horizontal parallax 

p = sun's horizontal parallax. 

s = semi-diameter of earth's shadow. 

I = inclination of relative orbit. 

h = moon's hourly motion on relative orbit. 

T, M, r?, m, A, d, 6, P, and p, are derived from Tables of the sun and moon. 
(See Problems IX and XIV.) 

The quantities s, I, and h, may be determined from these : 

3 = P + p — <5 + ,V (P + p — i) (441 and 445) .. . (89) ; 

tangI = M=nsr (449) ••• (90) * 



168 ECLIPSES OF THE SUN AND MOON. 

The triangle C'Mm' gives 



C'M = 



cos M Cm 



M — m 

? OT > k = T ' ' ' ( 91 )' 



COS I 



451. The above quantities being supposed to be known, let N'CF (Fig. 74) re- 
present the ecliptic, and C the stationary centre of the earth's shadow. Let 



Fig. 74. 



JLJi 









• 


P 


C (EC 



CC = A, and let N'C'L' represent the relative orbit of the moon. We here sup- 
pose the moon to be north of the ecliptic at the time of opposition, and near its 
ascending node : when it is south of the ecliptic A is to be laid ofF below N'CF, 
and when it is approaching either node, the relative orbit is inclined to the right. 
Let the circle KFK'R, described about the centre C, represent the section of the 
earth's shadow at the moon ; and let/, /', and g, gf, be the respective places of the 
moon's centre, at the beginning and end of the eclipse, and at the beginning and 
end of the total eclipse. C/ = C/ = s + d, and Cg = Cg' = s— d. Draw CM 
perpendicular to N'C'L', and M will represent the place of the moon's centre when 
nearest the centre of the shadow : it will also be its place at the middle of the 
eclipse ; for since C/ = Cf, and CM is perpendicular to N'C/', M/ = M/. 

452. Middle of the eclipse. — The time of opposition being known, that of the 
middle of the eclipse will become known when we have found the interval (x) em- 
ployed by the moon in passing from M to C. Now 

MC 
(expressed in parts of an hour) x = —j — ; 

and in the right-angled triangle CC'M we have CC = X, and < C'CM = 

< C'N'C = I, and therefore MC' = A sin I ; whence, by substitution, 

A sin I A sin I , nt A sin I cos I 

(equa. 91; = 



M 



M 



or, (expressed in seconds,) x = 



3600s. cos I 



M- 



A sin I 



Hence, if M = time of middle, we have 



M = T T x = T * 36 ^7 1 . Xsin I . . . (93) 

It is obvious that the upper sign is to be used when the latitude is increasing, 
and the lower sign when it is decreasing. 

The distance of the centre of the moon from the centre of the shadow at the mid- 
dle of the eclipse, 

= CM = CC'cosC'CM = AcosI . . . (94). 

453. Beginning and end of the eclipse. — Let any point I of the relative orbit be 
the place of the moon's centre at the time of any given phase of the eclipse. Let 
t = the interval of time between the given phase and the middle ; and k = C£, 



CALCULATION OF A LUNAR ECLIPSE. 169 

the distance of the centres of the moon and shadow. In the interval t the moon's 
centre will pass over the distance Ml ; hence 

t _ M[ = M/.cosI 
h M — m ' 

but, Ml = V ci 2 — CM^ = V tf — \2 cos2 I (equa. 94), 

and therefore t = C ° S * V ^ — X2 CO s2 I ; 

M — m 



3600s. cos I 



or, (in seconds,) t = V (k -+- X cos I) (& — X cos I) . . . (95) 

M' — m 

Let T' denote the time of the supposed phase of the eclipse, and M the time of 
the middle ; and we shall have 

T' = M -f t, or T = M — t, 
according as the phase follows or precedes the middle. 
Now, at the beginning and end of the eclipse, we have 
k=CforCf = «-4-<Z: 
substituting in equation (95) we obtain 

t' = 36 ^°^ C ^ S I \ / (s + d + X cos I) (a + d — X cos I) . . . (96). 

t' being found, the time of the beginning (B,) and the time of the end (E,) result 
from the equations 

B = M — t', E = M + t'. 
451. Beginning and end of the total eclipse. — At the beginning and end of the 
total eclipse, k = Cg = Cg' = s — d ; whence, by equation (95,) 



3600s. cos I 



t» = _^ s/ ( S _ d + X cos I) (s — d— \ cos I) ... (97) : 

M — m 

and, denoting the time of the beginning by B' and the time of the end by E', we 
have B = M — t", E' = M + t". 

455. Quantity of the eclipse. — In a partial eclipse of the moon the magnitude 
or quantity of the eclipse is measured by the relative portion of that diameter of 
the moon, which, if produced, would pass through the centre of the earlh's shad- 
ow, that is involved in the shadow. The whole diameter is divided into twelve 
equal parts, called Digits, and the quantity is expressed by the number of digits 
and fractions of a digit in the part immersed. When the moon passes entirely 
within the shadow, as in a total eclipse, the quantity of the eclipse is expressed by 
the number of digits contained in the part of the same diameter prolonged outward, 
which is comprised between the edge of the shadow and the inner edge of the moon. 
Thus the number of digits contained in SN (Fig. 74) expresses the quantity of the 
eclipse represented in the figure. Hence, if Q = the quantity of the eclipse, we 
shall have 

NS _ 12NS _ 12(NM + MS) _ 12 (NM + CS — CM) _ 
Q _ Xnv "~ "W ~ NV ~ NV ~~ 

12(<2-{-g_AcosI) t 
2d ' 

Q = 6( * + <*7 Xc ° -^...(98). 

If X cos I exceeds (s -f- d) there will be no eclipse. If it is intermediate between 
(* -j- d) an d ( s — d) there will be a partial eclipse ; and if it is less than (s — d) 
the eclipse will be total. 

CONSTRUCTION OF AN ECLIPSE OF THE MOON. 

456. The times of the different phases of an eclipse of the 
moon may easily be determined by a geometrical construction, 
within a minute or two of the truth. Draw a right line N 7 F 

22 



170 



ECLIPSES OF THE SUN AND MOON. 



(Fig. 75) to represent the ecliptic ; and assume upon it any 
point C, for the position of the centre of the earth's shadow at 
the time of opposition. Then, having fixed upon a scale of equal 

Fig. 75. 




parts, lay off CR = M — m, the difference of the hourly motions 
of the sun and moon in longitude ; and draw the perpendiculars 
CC' = X the moon's latitude in opposition, and RL' = X± n, the 
moon's latitude an hour after opposition. The right line C'L', 
drawn through C and L', will represent the moon's relative orbit. 
It should be observed, that if the Jatitudes are south they must be 
laid off below N'F, and that N'C'L' will be inclined to the right 
when the latitude is decreasing. With a radius CE =s (equation 
89) describe the circle EKFK', which will represent the section 
of the earth's shadow. With a radius = s -f- d, and another radius 
= s — d, describe about the centre C arcs intersecting N'L' in 
/,/', and g, g' ; / and/' will be the places of the moon's centre at 
the beginning and end of the eclipse, and g and g' the places at 
the beginning and end of the total eclipse. From the point C let 
fall upon N'C'L' the perpendicular CM ; and M will be the place 
of the moon's centre at the middle of the eclipse. To render the 
construction explicit, let us suppose the time of opposition to be 
7h. 23m. 15s. At this time the moon's centre will be at C. To 
find its place at 7h., state the proportion, 60m. : 23m. 15s. : : moon's 
hourly motion on the relative orbit : a fourth term. This fourth 
term will be the distance of the moon's centre from the point C at 
7 o'clock ; and if it be taken in the dividers and laid off on the 
relative orbit from C backward to the point 7, it will give the 
moon's place at that hour. This being found, take in the divi- 
ders the moon's hourly motion on the relative orbit, and lay it off 
repeatedly, both forward and backward, from the point 7, and 
the points marked off, 8, 9, 10, 6, 5, will be the moon's places at 
those hours respectively. Now, the object being to find the times 
at which the moon's centre is at the points/,/', g, g' , and M, let 



ECLIPSES OF THE SUN — LUMINOUS FRUSTUM. 



171 



the hour spaces thus found be divided into quarters, and these 
subdivided into 5-minute or minute spaces, and the times answer- 
ing to the points of division that fall nearest to these points, will 
be within a minute or so of the times in question. For example, 
the point/' falls between 9 and 10, and thus the end of the eclipse 
will occur somewhere between 9 and 10 o'clock. To find the num- 
ber of minutes after 9 at which it takes place, we have only to 
divide the space from 9 to 10 into four equal parts or 1 5-minute 
spaces, subdivide the part which contains f into three equal parts 
or 5-minute spaces, and again that one of these smaller parts 
within winch/' lies, into five equal parts or minute spaces. 

ECLIPSES OF THE SUN. 

457. An eclipse of the sun is caused by the interposition of the 
moon between the sun and earth ; whereby the whole, or part of 
the sun's light, is prevented from falling upon certain parts of the 
earth's surface. 

Let AGB and agb (Fig. 76) be sections of the sun and earth 



Fig. 76. 




by a plane passing through their centres S and E, Ac, Bb tan- 
gents to the circles AGB and agb on the same side, and Ad, Be 
tangents to the same on opposite sides. The figure AabB will be 
a section through the axis, of a frustum of a cone formed by rays 
tangent to the sun and earth on the same side, and the triangular 
space Fed will be a section of a cone formed by rays tangent on 
opposite sides. An eclipse of the sun will take place somewhere 
upon the earth's surface, whenever the moon comes within the 
frustum AabB, and a total or an annular eclipse whenever the 
moon comes within the cone Fed. 

458. Let mm'M (Fig. 76) be a circular arc described about the 
centre E, and with a radius equal to the distance of the centres 
of the moon and earth at the time of conjunction. The angle 
mES is the apparent semi-diameter of a section of the frustum, 
and m'ES the apparent semi-diameter of a section of the cone, at 
the distance of the moon. To find expressions for these semi- 
diameters in terms of determinate quantities, let the first be de- 
noted by A, and the second by A' ; and let P = the parallax of 



172 ECLIPSES OF THE SUN AND MOON. 

the moon, p = the parallax of the sun, and 8 = the semi-diameter 
of the sun. Then we have 

mES = A = mEA + AES = Erna - EA.m + AES ; 
or, A = P — p -f $ . . . (99) : 

and w'ES = m'EB - BES = Em'c - EBm' — BES ; 
or, A' = P —p-i . . . (100). 

Taking the mean values of P, p, and 6, (441,) we find for the 
mean value of A 1° 12' 53", and for the mean value of A' 40' 51". 

459. As the plane of the moon's orbit is not coincident with 
the plane of the ecliptic, an eclipse of the sun can happen only 
when conjunction or new moon takes place in one of the nodes 
of the moon's orbit, or so near it that the moon's latitude does not 
exceed the sum of the semi-diameters of the moon and of the lu- 
minous frustum (457) at the moon's orbit. This may be illustrated 
by means of Fig. 72, already used for a lunar eclipse, by supposing 
the sun to be in the directions Es, Es', Es", and that s, s' , s", are 
sections of the luminous frustum corresponding to these directions 
of the sun, also that ??i, m', m", represent the moon in the cor- 
responding positions of conjunction. Thus, denoting the moon's 
semi-diameter by d, and the greatest latitude of the moon in con- 
junction, at which an eclipse can take place, by L, we have 

L =P -p + 8 + d . . . (101). 
For a total eclipse, the greatest latitude will be equal to the sum 
of the semi-diameters of the moon and the luminous cone. Hence, 
denoting it by 1/, 

L' = P— p -J + iT.., (102). 
In order that an annular eclipse may take place, the apparent 
s e mi-diameter of the moon must be less than that of the sun, and 
the moon must come at conjunction entirely within the luminous 
frustum. Whence, if L" = the maximum latitude at winch an 
annular eclipse is possible, we have 

L" = P-j»+.* - d . . . (103). 

460. In the same manner as in the case of an eclipse of the 
moon, it has been found that when at the time of mean new moon 
the difference of the mean longitudes of the sun or moon and of 
the node, exceeds 19° 44', there cannot be an eclipse of the sun; 
but when the difference is less than 13° 33', there must be one. 
These numbers are called the Solar Ecliptic Limits. 

461. In order to discover at what new moons in the course of a 
year an eclipse of the sun will happen, with its approximate time, 
we have only to find the mean longitudes of the sun and node at 
each mean new moon throughout the year, (418,) and take the 
difference of the longitudes and compare it with the solar ecliptic 
limits. (For a more direct method of solving this problem, see 
Prob. XXVIII.) 

462. Eclipses both of the sun and moon recur in nearly the 



NUMBEJl OF ECLIPSES IN A YEAR, 173 

Cant? order and at the same intervals at the expiration of a period 
of 223 lunations, or 18 years of 365 days, and 15 days;* which 
for this reason is called the Period of the Eclipses. For, the 
time of a revolution of the sun with respect to the moon's node is 
346.61985 Id., and the time of a synodic revolution of the moon 
is 29.5305887d. These numbers are very nearly in the ratio of 
223 to 19. Thus, in a period of 223 lunations, the sun will have 
returned 19 times to the same position with respect to the moon's 
node, and at the expiration of this period will be in the same posi- 
tion with respect to the moon and node as at its commencement. 
The eclipses which occur during one such period being noted, 
subsequent eclipses are easily predicted. 

This period was known to the Chaldeans and Egyptians, by 
whom it was called Saros. 

463.* As the solar ecliptic limits are more extended than the lu- 
nar., eclipses of the sun must occur more frequently than eclipses 
of the moon. 

As to the number of eclipses of both luminaries, there cannot be 
fewer than two nor more than seven in one year. The most usual 
number is four, and it is rare to have more than six. When there 
are seven eclipses in a year, five are of the sun and two of the 
moon ; and when but two, both are of the sun. The reason is ob- 
vious. The sun passes by both nodes of the moon's orbit but once 
in a year, unless he passes by one of them in the beginning of the 
year, in which case he will pass by the same again a little before 
the end of the year, as he returns to the same node in a period of 
346 days. Now, if the sun be at a little less distance than 19° 44' 
from either node at the time of mean new moon, he may be eclipsed 
(450), and at the subsequent opposition the moon will be eclipsec 
near the other node, and come round to the next conjunction before 
the sun is 13° 33' from the former node : and when three eclipses 
happen about either node, the like number commonly happens 
about the opposite one ; as the sun comes to it in 173 days after 
wards, and six lunations contain only four days more. Thus there 
may be two eclipses of the sun and one of the moon about each of 
the nodes ; and the twelfth lunation from the eclipse in the begin 
ning of the year may give a new moon before the year is ended, 
which, in consequence of the retrogradation of the nodes, may be 
within the solar ecliptic limit ; and hence there may be seven 
eclipses in a year, five of the sun and two of the moon. But when 
the moon changes in either of the nodes, she cannot be near enough 
to the other node, at the next full moon, to be eclipsed, as in the 
interval the sun will move over an arc of 14° 32', whereas the 
greatest lunar ecliptic limit is but 13° 21', and in six lunar months 
afterwards she will change near the other node ; in this case there 
cannot be more than two eclipses in a year, both of which will be 

* More exactly, 18 years (of 365 days) plus 15c 7h. 42m. 29s. 



174 ECLIPSES OF THE SUN AND MOON. 

of the sun. If the moon changes at the distance of a few degrees 
from either node, then an eclipse hoth of the sun and moon will 
probably occur in the passage of that node and also of the other. 

464. Although solar eclipses are more frequent than lunar, when 
considered with respect to the whole earth, yet at any given place 
more lunar than solar eclipses are seen. The reason of this cir- 
cumstance is, that an eclipse of the sun (unlike an eclipse of the 
moon) is visible only over a part of a hemisphere of the earth. To 
show this, suppose two lines to be drawn from the centre of the 
moon tangent to the earth at opposite points : they will make an 
angle with each other equal to double the moon's horizontal paral- 
lax, or of 1° 54'. Therefore, should an observer situated at one 
of the points of tangency, refer the centre of the moon to the cen- 
tre of the sun, an observer at the other would see the centres of 
these bodies distant from each other at an angle of 1° 54', and their 
nearest limbs separated by an arc of more than 1°. 

465. Instead of regarding an eclipse of the sun as produced by 
an interposition of the moon between the sun and earth, as we have 
hitherto considered it, we may regard it as occasioned by the moon's 
shadow falling upon the earth. Fig. 77 represents the moon's 
shadow, as projected from the sun and covering a portion of the 
earth's surface. Wherever the umbra falls, there is a total eclipse ; 
and wherever the penumbra falls, a partial eclipse. 

Fig. 77. 




466. In order to discover the extent of the portion of the earth's 
surface over which the eclipse is visible at any particular time, 
we have only to find the breadth of the portion of the earth covered 
by the penumbral shadow of the moon ; but we will first ascertain 
the length of the moon's shadow. As seen at the vertex of the 
moon's shadow, the apparent diameters of the moon and sun are 
equal. Now, as seen at the centre of the earth, they are nearly 
equal, sometimes the one being a little greater and sometimes the 
other. It follows, therefore, that the length of the mooris shadow 
is about equal to the distance of the earth, being sometimes a little 
greater and at other times a little less. 



175 

When the apparent diameter of the moon is the greater, the 
shadow will extend beyond the earth's centre ; and when the ap- 
parent diameter of the sun is the greater, it will fall short of it. If 
we increase the mean apparent diameter of the moon as seen from 
the earth's centre, viz. 31' 7", by ¥ V> the ratio of the radius of the 
earth to the distance of the moon, we shall have 31' 38" for the 
mean apparent diameter of the moon as seen from the nearest point 
of the earth's surface. Comparing this with the mean apparent 
diameter of the sun as viewed from the same point, which is sen- 
sibly the same as at the centre of the earth, or 32' 2", we perceive 
that it is less ; from which we conclude, that when the sun and 
moon are each at their mean distance from the earth, the shadow 
of the moon does not extend as far as the earth's surface. 

467. To find a general expression for the length of the moon's 
shadow, let AGB, a'g'b', and agb (Fig. 78) be sections of the sun, 

Fig. 78. 



moon, and earth, by a plane passing through their centres S, M, 
and E, supposed to be in the same right line, and Aa', Bb' tan- 
gents to the circles AGB, a'g'b' : then a'Kb' will represent the 
moon's shadow. Let L — the length of the shadow ; D = the dis- 
tance of the moon ; D' = the distance of the sun ; d = the appa- 
rent semi-diameter of the moon ; and 5 = apparent semi-diameter 
of the sun. At K the vertex of the shadow, MKa' the apparent 
semi-diameter of the moon, will be equal to SKA the apparent se- 
mi-diameter of the sun ; and as the distance of this point from the 
centre of the earth, even when it is the greatest, is small in com- 
parison with the distance of the sun (466), the apparent semi-diam- 
eter of the sun will always be very nearly the same to an observer 
situated at K as to one situated at the centre of the earth. Now, 
since the apparent semi-diameter of the moon is inversely propor- 
tional to its distance, 

angle MKa' : d : : ME : MK ; 
and thus, <5 : d : : ME : MK : : D : L (nearly) : 

whence, L==D 7 * * : ( 104 )* 

If a more accurate result be desired, we have only to repeat the cal-: 
culations, after having diminished 8 in the ratio of D' to (D'+L—D). 

468. Now, to find the breadth of the portion of the earth's surface covered by 
the penumbral shadow, let the lines Ad', Be' (Fig. 78) be drawn tangent to the 
circles AGB, a'g'b', on opposite sides, and prolonged on to the earth. The space 



176 ECLIPSES OF THE SUN AND MOON. 

hdd'k will represent the penumbra of the moon's shadow, and the arc gh one half 
the breadth of the portion of the earth's surface covered by it. Let this arc or the 
angle gEh = S, and denote the semi-diameter of the sun and the semi-diamete; 
and parallax of the moon by the same letters as in previous articles. The triangle 
MEA gives 

angle MEA = S = MAZ — AME. 

The angle AME is the moon's parallax in altitude at the station A, and MAZ is 
its zenith distance at the same station. Denote the former by P' and the latter by Z, 
Thus, S = Z — F . . . (105). 

The triangle AMS gives 

AME = F = MSA -f MAS; 
MAS= d-{-6 ; and MSA is the sun's parallax in altitude at the station A: let it 
be denoted by p . We have, then, 

Y = d + 6+p' = d + 6 (nearly) . . . (106); 
and to find Z we have (equa. 9, p. 51), 

P 
P ' = P sin Z, or sin Z = — . . . (107). 

P' and Z being found by these equations, equa. (105) will then make known the 
value of S. 

If great accuracy is required, the calculation must be repeated, giving now to 
pi in equation (106) the value furnished by equation '9) which expresses the rela- 
tion between the parallax in altitude of a body and its horizontal parallax, instead 
of neglecting it as before ; and Z must be computed from the following equation : 

siaZ = !i^|:. . . (103). 
sin P 

The penumbral shadow will obviously attain to its greatest 
breadth when the sun is in its perigee and the moon is in its apo- 
gee. The values of d, 6, and P under these circumstances are re- 
spectively 14' 41", 16' 18'', and 53' 48". Performing the calcula- 
tions, we find that the breadth of the greatest portion of the earth's 
surface ever covered by the penumbral shadow is 69° 18', or about 
4800 miles. 

469. The breadth of the spot comprehended within the umbra 
mav be found in a similar manner. 

The arc gh' (Fig. 78) represents one half of it : denote this arc or the equal an- 
gle gEh' by S'. 

MEA' = S' = MA'Z' — A*ME ; 
or, S' = Z— F . . . (109). 

i'ME = P' = MSA' + MA'S ; 
but MA'S = d — <5, and MSA' = p', sun's parallax in altitude at A' ; whence, 

P' = d — c + p' = d—c (nearly) . . . (110) : 
and wehai€, as before, 

F 
P' = PsinZ,or sinZ=— . . . (111). 

The greatest breadth will obtain when the sun is in its apogee 
and the moon is in its perigee. We shall then have 

6 = 15' 45", d = 16' 45", P = 61' 24". 

Making use of these numbers, we deduce for the maximum 
breadth of the portion of the earth's suiface covered by the moons 
shadow, 1° 50', or 127 miles. 

470. It should be observed that the deductions of the last two 



CALCULATION OP AN ECLIPSE OF THE SUtf. 177 

articles answer to the supposition that the moon is in the node, and 
that the axis of the shadow and penumbra passes through the cen- 
tre of the earth. In every other case, both the shadow and pe- 
numbra will be cut obliquely .by the earth's surface, and the sec- 
tions will be ovals, and very nearly true ellipses, the lengths of 
which may materially exceed the above determinations. 

471. Parallax not only causes the eclipse to be visible at some 
places and invisible at others, as shown in Art. 464 ; but, by making 
the distance of the centres of the sun and moon unequal, renders 
the circumstances of the eclipse at those places where it is visible 
different at each place. This may also be inferred from the cir- 
cumstance that the different places, covered at any time by the 
shadow of the moon, will be differently situated within this shadow. 
It will be seen, therefore, that an eclipse of the sun has to be con- 
sidered in two points of view: 1st. With respect to the whole 
earth, or as a general eclipse ; and, 2d. With respect to a particu- 
lar place. 

472. The following are the principal facts relative to eclipses of the sun that 
remain to be noticed : 1st. The duration of a general eclipse of the sun cannot ex- 
ceed about 6 hours. 2d. A solar eclipse does not happen at the same time at all 
places where it is seen : as the motion of the moon beyond the sun, and conse- 
quently of its shadow, is from west to east, the eclipse must begin earlier at the 
western parts and later at the eastern. 3d. The moon's shadow being tangent to 
the earth at the commencement and end of the eclipse, the sun will be just rising 
at the place where the eclipse is first seen, and just setting at the place where it is 
last seen. At the intermediate places, the sun will at the time of the beginning and 
end of the eclipse have various altitudes. 4th. An eclipse of the sun begins on the 
western side and ends on the eastern. 5th. When the straight line passing through 
the centres of the sun and moon passes also through the place of the spectator, the 
eclipse is said to be central : a central eclipse may be either annular or total, ac- 
cording as the apparent diameter of the sun is greater than that of the moon, or 
the reverse. 6th. A total eclipse of the sun cannot last at any one place more than 
eight minutes ; and an annular eclipse more than twelve and a half minutes. 7th, 
In most solar eclipses the moon's disc is covered with a faint light, a phenomenon 
which is attributed to the reflection of the light from the illuminated part of the 
earth. 

CALCULATION OF AN ECLIPSE OF THE SUN. 

(1.) Of the circumstances of the general eclipse. 

473. It is a simple inference from what has been established in Art. 459, that an 
eclipse of the sun will begin and end upon the earth, at the times before and after 
conjunction, when the distance of the centres of the moon and sun is equal to 
P — p-\-S-{-d', that the total eclipse will begin and end when this distance is 
equal to P — p — <$-f-<2; and the annular eclipse when the distance is equal to 
P— J> + <5 — d. 

474. The times of the various phases of the general eclipse of the sun may be 
obtained by a process precisely analogous to that by which the times of the pL.nses 
of an eclipse of the moon are found. Let C (Fig. 79) be the centre of the sun, and 
C the centre of the moon, at the time of conjunction. We may suppose the sun 
to remain stationary at C, if we attribute to the moon a motion equal to its mo- 
tion relative to the sun ; for, on this supposition, the distance of the centres of the 
two bodies will, at any given period during the eclipse, be the same as that which 
obtains in the actual state of the case. Let N'C'L' represent the orbit that would 
be described by the moon if it had such a motion, which is called the Relative Or- 
bit. Let CM be drawn perpendicular to it; and let C/= C/' = P — p-M + ^. 
and Cg = Cg 1 =P — p — 6-\-d, or P — p + $ — d, according as the eclipse is to- 

23 



178 ECLIPSES OF THE SUN AND MOON. ' 

tal or annular. Then, M will be the place of the moon's centre at the middle of 
the eclipse ; f&ndf the places at the beginning and end of the eclipse ; andg - and 
g the places at the beginning and end of the total, or of the annular eclipse. We 
shall thus have, as in eclipses of the moon, 

Fig. 79. 




tang I = ^ , CM = X cos 1, CM = A sin I . . . (112). 

5 M — m 

3600s. X sin I cos I 
interval from con. to mid. = ^ . . . (113). 

1V1 777.. 

Interval from middle to beginning or end 
3600 s. cos I 

= M 771 

Interval for total eclipse 

3600s. cos I 
~~ M — m 
Interval for annular eclipse 

3600s. cos I 



v/(A-' + X cos !)(*' — Xcos I) . . . (114). 



>/(&" + Xcos I) (#/_ a cos I) . . . (115). 



M — 



J(k"'+\ cos I) (*"' — X cos I) . . . (116). 



6(&' — Xcos I) 
Quantity = . . . (117). 

k'=.P—p + 6 + d,k"=Y—p — S + d,k!" = 'P—p + S—d . . . (118). 

The letters X, M, tti, &c, represent quantities of the same name as in the formulae 
for a lunar eclipse ; but they designate the values of these quantities at the time of 
conjunction, instead of opposition. These values are in practice obtained from ta- 
bles of the sun and moon, as in a lunar eclipse. 

475. The times of the different circumstances of a general eclipse of the sun 
may also be found within a minute or two of the truth, by construction, in a pre- 
cisely similar manner with those of an eclipse of the moon, (456.) 

(2.) Of the phases of the eclipse at a particular place. 

476. The phase of the eclipse, which obtains at any instant at a given place, is 
indicated by the relation between the apparent distance of the centres of the sun 
and moon, and the sum, or difference, of their apparent semi-diameters : and the 
calculation of the time of any given phase of the eclipse, consists in the calculation 
of the time when the apparent distance of the centres has the value relative to the 
sum or difference of the semi-diameters, answering to the given phase. Thus, if 
we wish to find the time of the beginning of the eclipse, we have to seek the time 
when the apparent distance of the centres of the sun and moon first becomes equal 
to the sum of their apparent semi-diameters. 

477. The calculation of the different phases of an eclipse of the sun, for a par- 
ticular place, involves, then, the determination of the apparent distance of the cen- 
tres of the sun and moon, and of the apparent semi-diameters of the two bodies, 
at certain stated periods. 

The true semi-diameter of the sun, as given by the tables, may be taken for the 
apparent without material error. For the method of computiug the apparent semi- 
diameter of the moon, for any given time and place, see Problem XVII. 



SOLAR ECLIPSE. APPROXIMATE TIMES OF PHASES. 179 

478. According to the celebrated astronomer Dusejour, in order to make the ob- 
servations agree with theory, it is necessary to diminish the sun's semi-diameter, as 
it is given by the tables, 3 ".5. This circumstance is explained by supposing that 
the apparent diameter of the sun is amplified, by reason of the very lively impres- 
sion which its light makes upon the eye. This amplification is called Irradiation. 
He also thinks that the semi-diameter of the moon ought to be diminished 2", to 
make allowance for an Inflexion of the light which passes near the border of this 
luminary, supposed to be produced by its atmosphere. It must be observed, how- 
ever, that the astrouomei's of the present day do not agree either as to the neces- 
sitv or the amount of the diminutions just spoken of. 

47°. The determination of the apparent distance of the centres of the sun and 
moon may easily be accomplished, as will be shown in the sequel, when the ap- 
parent longitude and latitude of the two bodies have been found. Now, the true 
longitude of the sun, and the true longitude and latitude of the moon, may be found 
from the tables, (Probs. IX and XIV) ; and from these the apparent longitudes and 
latitudes may be deduced by correcting for the parallax. But the customary mode 
of proceeding is a little different from this: the true iongitude and latitude of the 
sun are employed instead of the apparent, and the parallax of the sun is referred to 
the moon ; that is. the difference between the parallax of the moon and that of the 
sun is, by fiction, taken as the parallax of the mGon. This supposititious parallax is 
called the moon's Relative Parallax. (See Prob. XVII.) 

480. We will first show how to find the approximate times of the different phases 
of the eclipse. Put T = the time of new moon, known to within 5 or 10 minutes. 
(Prob. XXVII.) For the time T calculate by the tables the sun's longitude, hourly 
motion, and semi-diameter, and the moon's longitude, latitude, horizontal parallax, 
semi-diameter, and hourly motions in longitude and latitude. Subtract the sun's 
horizontal parallax from the reduced horizontal parallax of the moon,* and calcu- 
late the apparent longitude and latitude, and the apparent semi-diameter of the 
moon. From a comparison of the apparent longitude cf the moon with the true 
longitude of the sun, we shall know whether apparent ecliptic conjunction occurs 
before or after the time T. Let T' denote the time an hour earlier or later than 
the time T, according as the apparent conjunction is earlier or later. With the 
sun and moon's longitudes, the moon's latitude, and the hourly motions in longi- 
tude and latitude, at the time T, calculate the longitudes and the moon's latitude 
for the time T' ; and for this time also calculate the moon's apparent longitude 
and latitude. Take the difference between the apparent longitude of the moon and 
the true longitude of the sun at the time T, and it will be the apparent distance 
of the moon from the sun in longitude, at this time. Let it be denoted by n. Find, 
in like manner, the apparent distance of the moon from the sun in longitude at the 
time T', and denote it by n . In the same manner as at the time T, we find wheth- 
er apparent conjunction occurs before or after the time T'. If it occurs between 
the times T and T', the sum of n and n', otherwise their difference, will be the 
apparent relative motion cf the sun and moon in longitude in the interval T' — T, 
or T — T' ; from which the relative hourly motion will become known. The dif- 
ference of the apparent latitudes of the moon, at the times T and T', will make 
known the apparent relative hourly motion in latitude. With the relative hourly 
motion in longitude and the difference of the apparent longitudes at the time T, 
find by simple proportion the interval between the time T and the time of apparent 
ecliptic conjunction ; and then, with the apparent latitude of the moon at the time 
T and its hourly motion in latitude, find the apparent latitude at the time of ap- 
parent conjunction thus determined. Then, knowing the relative hourly motion 
of the sun and moon in longitude and latitude, together with the time of apparent 
conjunction, and the apparent latitude at that time, and regarding the apparent 
relative orbit of the moon as a right line, (which it is nearly,) it is plain that the 
time of beginning, greatest obscuration, and end, as well as the quantity of the 
eclipse, may be calculated after the same manner as in the general eclipse ; the 
disc of the sun answering to the section of the luminous frustum mentioned in Art 



* The reduced horizontal parallax of the moon is its horizontal parallax as re 
duced from the equator to the given place. (See Prob. XV.) 



180 



ECLIPSES OF THE SUN AND MOON, 




457, and the apparent element* 
answering to the true. Let C 
(Fig. 80) represent the centre 
of the sun supposed stationary, 
CC the apparent latitude of the 
moon at apparent conjunction, 
If'O' the apparent relative orbit 
of the moon, determined by its 
passing through the point C 
and making a determinate an- 
gle with the ecliptic N'P, or by 
its passing through th« situa- 
tions of the moon at the times 
T and T'. Also, let RKFK' 
be the border of the sun's disc;. 
/,/' the positions of the moon's 
centre at the beginning and end of the eclipse, determined by describing a circle 
around C as a centre, with a radius equal to the sum of the apparent semi-diame- 
ters of the sun and moon ; and M (the foot of the perpendicular let fall from €X 
upon N'C) its position at the time of greatest obscuration. 

If the eclipse should be total or annular, then g, g' will be the positions of the 
moon's centre at the beginning and end of the total or annular eclipse ; these 
points being determined by describing a circle around C as a centre, and with a 
radius equal to the difference of the apparent semi-diameters of the sun and 
moon. 

The results will be a closer approximation to the truth, if the same calculations 
that are made for the time T* be made also for another time T". 

The various circumstances of the eclipse may also be had by construction, after 
the same manner as in a lunar eclipse, (456.) 

481. In order to be able to observe the beginning or end of a solar eclipse, it is 
necessary to know the position of the point on the sun's limb where the first or 
Fast contact takes place. The situation of these points is designated by the dis- 
tance on the limb, intercepted between them and the highest point of the limb, call- 
ed the Vertex. The contacts will take place at the points t, f, (Fig. 80,) on the 
lines Cf, Cf. To find the position of the vertex, with the sun's longitude found 
for the beginning of the eclipse, calculate the angle of position of the sun at that 
time, (see Prob. XIII,) and lay it off to the right of the circle of latitude CK when 
the sun's longitude is between 90° and 270°, and to the left when the longitude is 
less than 90° or more than 270°. Suppose CF to be the circle of declination thus 
determined. Next, let Z (Fig. 24, p. 47) be the zenith, P the elevated pole, and & 
the sun ; then in the triangle ZPS we shall know ZP the co-latitude, ZPS the hour 
angle of the sun, and we may deduce PS, the co-declination of the sun, from the 
longitude of the sun as derived from the tables, (equa. 35.) These three quanti- 
ties being known, ZSP, the angle made by the vertical through the sun with its 
circle of declination, may be computed ; and being laid off in the figure to the 
right or left of CP, (Fig. 80,) according as the time of beginning is before or after 
noon, the point Z or Z', as the case may be, in which the vertical intersects the 
limb RKK', will be the vertex, and the arc Zt, or Z't, on the limb, will ascertain 
the situation of t, the first point of contact, with respect to it. 

The situation of the last point of contact may be found by the same mode of 
proceeding. 

482. Let us now show how to find the exact times of the beginning, greatest 
obscuration, and end of the eclipse, the approximate times being known. Let B 
designate the approximate time of beginning, taken to the nearest minute. Cal- 
culate for the time B by means of the tables, the sun's longitude, hourly motion, 
and semi-diameter ; also the moon's longitude, latitude, horizontal parallax, semi- 
diameter, and hourly motions in longitude and latitude. Then, making use of thc« 
relative parallax, calculate the apparent longitude, latitude, and semi-diameter of 
the moon. Subtract the apparent longitude of the moon from the true longitud* 
of the sun ; the difference wiH be the apparent distance of the moon from the sun 
in longitude : let it be denoteAjby o. Denote the apparent latitude of the moor 
byX. 




SOLAR ECLIPSE — TRUE TIMES OF PHASES. 181 

How, let EC (Fig 81) represent an arc of the ecliptic, pj ff> g^ 

•and K its pole ; and let S be the situation of the sun, 
and M the apparent situation of the moon at the time B. 
Then MS is the apparent distance of the centres of the 
two bodies at this time. Denote it by A. Sm = a, 
and Mm = A. The right-angled triangle MSm being 
very small, may be considered as a plane triangle, and 
we therefore have, to determine A, the equation 
A3 = «2 + x2 . . . (119) * 

483. Having computed the value of a, we find, by 
■comparing it with the sum of the apparent semi-diame- 
iers of the sun and moon, whether the beginning of the 
^eclipse occurs before or after the approximate time B. Fix 
upon a time some 4 or 5 minutes before or after B, ac- .g- 
«ording as the beginning is before or after, and call it B'. 
With the sun and moon's longitudes, the moon's latitude, and the hourly motions 
in longitude and latitude, at the time B, find the longitudes and the moon's lati- 
tude at the time B', and compute for this time the apparent longitude, latitude, 
and semi-diameter of the moon. Subtract the apparent longitude of the moon 
from the true longitude of the sun, and we shall have the apparent distance of the 
moon from the sun at the time B'. Take the difference between this and the same 
distance a at the time B, and we shall have the apparent relative motion of the 
sun and moon in longitude during the interval of time between B and B'. Then 
find, by simple proportion, the apparent relative hourly motion in longitude, and 
denote it by k~ Take the difference between the apparent latitudes of the moon 
at the times B and B', and it will be the apparent relative motion of the sun and 
moon in latitude, in the interval ; from which deduce the apparent relative hourly 
motion in latitude, and call it n. Now, put t = the interval between the ap- 
proximate and true times of the beginning of the eclipse, and suppose S and M 
(Fig. 81) to be the situations of the sun and moon at the true time of beginning. 
In the time t, the apparent relative motions in longitude and latitude wiH be, re- 
spectively, equal to kt and nt, and accordingly we shall have 

Sm = a — kt, Mm = X + nt. 

The small right-angled triangle SMm may be considered as a plane triangle ; the 
hypothenuse SM = rp = the sum of the apparent semi-diameters of the sun and 
moon, minus 5".5, (478.) We have then, to find t, the equation 

(a _H)2 4-(A + H *) 2 = ^ s 
•or, developing and transposing, 

(7*2 + A2) *2_ 2 (ak — \n)t = ^— («2+ X2) = <f2_ A 2 ; 
making A = if>2 _ a2, and B = ak — Xn, (n2 -j- &2) *2 _ 2Bt = A, 

and (= b-Maw^ 

n* + k* 

The negative sign must be prefixed to the radical, for, if we suppose A to be equal 
to zero, t must be equal to zero. Multiplying the numerator and denominator by 
B+ V B2 4r A (n%-\-k2), and restoring the value of A, we obtain 

3600s. (a2 — 4,2) 

(in seconds) t= - — — M2.N 

B+V B2+(f2_ A 2) (n2 + k2) ' ' ' ^ J ' 

Although this equation has been investigated for the beginning of the eclipse, it 
is plain that it will answer equally well for the determination of the other phases, 

* In place of equation (119) the following equations may be employed inloga^ 
rithmic computation: 

A a 

tang0 = - A = ; 

b a cos $ ' 
where B is an auxiliary arc 



182 



ECLIPSES OF THE SUN AND MOON. 



if we give the proper values and signs to \p, a, >, n, and Jc. k is positive before' 
conjunction and negative after it, and the radical quantity is negative after con* 
junction ; n is negative, when the moon appears to recede from the north pole of 
the ecliptic ; A h?.-z the sign — , when it is south ; a is always positive.* 

The value of i taken with its sign is to be added to the time B. 

484. The values of the quantities a, ^, n T and A', are found for the other phases- 
after the same manner a;> for the beginning. 

To obtain the value of T - at the time of greatest obscuration, find the rela- 
tive motions in longitude and latitude, (A and n,) during some short interval near 
the middle of the eclipse, which is the approximate time of greatest obscuration ; 
then compute the inclination of the relative orbit by the equation 



tang I = - . 



(122.) (See equa..9l0): 



Fig. 82. 



after which i£ will result from the equation 

$ = A cos I . . . (123.) (See equa. 94). 

A is the moon's latitude at the time of apparent conjunction, which is easily cal- 
culated, by means of the values of A: and n, and the apparent longitude and lati- 
tude of the moon, found for some instant near the time of apparent conjunction. 

For the beginning and end of the total eclipse, we have, y = appar. semi-diam> 
of moon — appar. semi-diam. of sun -f- l".o ; and for the beginning and end of the 
annular eclipse, ^ =^ appar. scmi-diam. of son — appar. semi-diam. of moon — 1".5, 
485. If the value of ty, given by equation (123,) be substituted in equation 
(121.) this equation will make known the time of greatest obscuration; but this 
may be found more conveniently by a different process. Let NCF (Fig, S2) repre- 
sent a portion of the ecliptic, EML a portion 
of the relative orbit passed over about the 
time of greatest obscuration, C the stationa- 
ry position of the sun's centre, and M the 
place of the moon's centre at the instant of 
its nearest approach to C. Also, let a = CR 
the apparent distance of the moon from the* 
sun in longitude at the time of the nearest 
approach of the centres, V = RM the moon's 
y C » N apparent latitude at the same time, A = MA 

the apparent relative motion in longitude in some short interval about this time ? 
and n = kn the moon's apparent motion in latitude during the same interval. The 
right-angled triangles Mn£ and CMR are similar,, for their sides are respectively 
perpendicular to each other ; whence, 

MA : MR : : kn : CR ; 





and 



CR 



MR ^-, or, « = *? • .. Q24). 



If the moon's apparent latitude be found for the approximate time of greatest 
obscuration, and substituted for A' in equation (124,; this equation will give very 
nearly the apparent distance (a) of the, two bodies in longitude at the true time of 
greatest obscuration. With this, and the apparent distance at the approximate 
time of greatest obscuration, together with the relative apparent motion in longi- 
tude, the true time of greatest obscuration may be found nearly by simple propor- 
tion. A more accurate result may then be had by finding the moon's apparent 
latitude for the time obtained, substituting it for V in equation (124) and then re- 
peating the calculations. 

486. A simpler, though less accurate method than that already given, of find- 
ing the times of beginning and end of the total or annular eclipse, is to compute 
the half duration of the total or annular eclipse, and add it to, and subtract it from, 

* Developing the radical in equation (120,) and neglecting all the terms after 
ti«j eecondj as being very small, we obtain for the beginning and end of the eclip6e 
the mm* convenient formula 



1 = 



1S00s.(a2 — xpZ) 
J* 



OCCULTATIONS. 183 

the time of greatest obscuration. This interval may easily be determined, if we 
can find the rate of motion on the relative orbit, and the distance passed over by 
the moon's centre during the interval. Let g, g' (Fig. 82) be the places of the 
moon's centr.j at the instants of the two interior contacts, and M/i the distance 
passed over in some short interval (L). Let 6 = < 3I/?A- the complement of the 
inclination of the relative orbit, k = MAr, /;' = Mn, and t = half duration of total 
or annular eclipse. The triangles MnA-, CRM, give 

Art Jc 

Mn = " . , or hf = -fU . . (125) : 

sin Mnk sin 

RM X' 

and tang RCM = tang Mnk = — '—, or, tang 9 = — . . . (126). 

Finding the value of 9 by the last equation, and substituting it in equation (125), 
we obtain the value of k' ; and then, to find t, we have 

\>:L::Kg:Uort = 1 ^. 
Mg- = ^~Cg 2 — CM 2 = v^2_A2 (Art. 484) ; 

V f L V ^ — A-2 LV^+ A)^^ ) 

whence, t = -, = z — — . . . (120 

487. The apparent distance of the centres of the two bodies at the time of great- 
est obscuration being known, the quantity of the eclipse may be readily found. 
We have but to subtract the apparent distance from the sum of the apparent semi, 
diameters, and state the proportion, as the sun's apparent diameter : the remain- 
der : : 12 digits : the digits eclipsed. (For a more particular description of the 
method of calculating a solar eclipse, see Prob. XXX.) 



OCCULTATIONS. 

488. At all places upon the earth's surface, which at a given 
time have the moon in the horizon, its apparent place will differ 
from its true place, by the amount of its horizontal parallax. It 
follows, therefore, that a star will be eclipsed by the moon some- 
where upon the earth, in case its true distance from the moon's 
centre is less than the sum of the moon's semi-diameter and hori- 
zontal parallax. 

The greatest value of the moon's semi-diameter is 16' 45", and 
that of its horizontal parallax 61' 24". If we add the sum of these 
numbers to 5° 17' 34", the maximum latitude of the moon, we ob- 
tain as the result 6° 35' 43". It is then only the stars which have 
a latitude less than 6° 35' 43" that can experience an occultation 
from the moon. 

489. By considering the various situations of the stars liable to an occultation, 
taking the greatest and least values of the sum of the moon's semi-diameter and 
horizontal parallax, and allowing for the inequalities of the motions of the moon, 
it has been found, that, if at the time of the mean conjunction of the moon and 
a star, (that is, when the moon's mean longitude is the same with the longitude of 
the star.) their difference of latitude exceed 1° 37', there cannot be an occultation ; 
if the difference be less than 51', there must be an occultation somewhere on the 
earth ; and that between these limits there is a doubt, which can only be removed 
by the calculation of the moon's true place. 

490. The calculation of an occultation is very nearly the same as that of a solar 
eclipse. The only difference is in the data. The star has no diameter, parallax, 
or motion in longitude ; and as it is situated without the ecliptic, we have, in place 
of the latitude of the moon, employed in solar eclipses, the difference between 



184 OF THE PLANETS AND THEIR PHENOMENA. 

the latitude of the moon and that of the star, and in place of the difference be- 
tween the longitudes of the two bodies and their relative hourly motion in longi- 
tude, these quantities referred to an arc passing through the star and parallel to the 
ecliptic. Thus, if EC (Fig. 81) represent the ecliptic, K its pole, s t!»e situation 
of the star, M that of the moon, and sm' an arc passing through s ami parallel to 
the arc EC, we have in place of mM, mTVI = mM — mm', and in place of Sm, sm'. 
The hourly variation of Sm must also be reduced to the arc sm'. 

Fig. 83. 491. The reduction of the difference of longitude of the 

^ moon and star, to the parallel to the ecliptic, passing 
through the star, is effected by multiplying this difference 
by the cosine of the latitude of the star. For, let AB (Fig. 
83) be an arc of the ecliptic, and A'B' the corresponding 
arc of a circle parallel to it ; then, since similar arcs of cir- 
cles are proportional to their radii, we have 

BC : BC : : AB : A'B' = AB ^'°' . 

B'C = Ca = B'C cos BCB' = BC cos BB* : 



AB.BCcosBB' . _. __. 

A'B' = z^ = AB cos BB'. 

BC 

The reduction of the relative hourly motion in longitude to the parallel in ques. 

tion, is obviously effected in the same manner. 




CHAPTER XVI. 

OF THE PLANETS, AND THE PHENOMENA OCCASIONED BY THEIR 

MOTIONS IN SPACE. 

APPARENT MOTIONS OF THE PLANETS WITH RESPECT TO 

THE SUN. 

492. The apparent motion of an inferior planet, with reference 
to the sun, is materially different from that of a superior planet. 
The inferior planets always accompany the sun, being seen alter- 
nately on the east and west side of him, and never receding from 
him beyond a certain distance, while the superior planets are seen 
at every variety of angular distance. This difference of apparent 
motion arises from the difference of situation of the orbits of an 
inferior and superior planet, with respect to the orbit of the earth ; 
the one lying within and the other without the earth's orbit. 

Let CAC'B (Fig. 84) represent the orbit of either one of the in- 
ferior planets, Venus for example, and PKT the orbit of the earth ; 
which we will suppose to be circles, and to lie in the same plane ; 
and let MLN represent the sphere of the heavens to which all bo 
dies are referred. Suppose, for the present, that the earth is sta 
tionary in the position P, and through P draw the lines PA, PB ; 
tangent to the orbit of Venus, and prolong them on till they inter 



APPARENT MOTIONS OF THE PLANETS. 



185 



sect the hoavens at a and b. When Venus is at C, (the earth be- 
ing at P,) she will be in superior conjunction, and when at C in 
inferior conjunction. Now, by inspecting the figure, it will be seen 
that in passing from C to C she will be seen in the heavens on the 
east side of the sun, and in passing from C to C on the west side 

Fig. 84. 




of the sun ; also, that in passing from C to A she will recede from 
the sun in the heavens, from A to C approach him, from C to B 
recede from him again, and from B to C approach him again, a 
and b will be her positions in the heavens at the times of her great- 
est eastern and western elongations. 

When Venus is to the east of the sun, she is seen in the even- 
ing, and called the Evening Star ; and when to the west, she is 
seen in the morning, and called the Morning Star. 

493. We have in the foregoing investigation supposed the earth 
to be stationary, a supposition which is contrary to the fact ; but 
it is plain that the only effect of the earth's motion in the case un- 
der consideration, as it is slower than that of the planet, is to cause 
the points A, C, B to advance in the orbit, without altering the 
nature of the apparent motion of the planet with respect to the sun. 
The orbits of the earth and planet are also ellipses of small eccen- 
tricity, and are slightly inclined to each other, instead of being cir- 
cles and lying in the same plane : on this account, as the greatest 
elongations will occur in various parts of the orbits, they will differ 
in value. The greatest elongation of Venus varies from 45° to 47° 
12'. Its mean value is about 46°. 

494. Owing to the circumstance of the orbit of Mercury being 

24 



186 OF THE PLANETS AND THEIR PHENOMENA. 

within the orbit of Venus, the greatest elongation of this planet is 
less than that of Venus. It varies between the limits 16° 12', and 
28° 48' ; and is, at a mean, 22° 30'. 

495. Next, suppose PKT (Fig. 84) to be the orbit of a superior 
planet, and CAC'B that of the earth ; and, as the velocity of the 
earth is much greater than that of the planet, let us, for the present, 
regard the planet as stationary in the position P, while the earth 
describes the circle CAC. When the earth is at C, the planet, 
being at P, is in conjunction with the sun. When the earth is at 
A, SAP, the elongation of the planet is 90°. When it arrives at 
C, the planet is in opposition, or 180° distant from the sun : and 
when it reaches B, the elongation is again 90°. At intermediate 
points the elongation will have intermediate values. If, now, we re- 
store to the planet its orbitual motion, we shall manifestly be con- 
ducted to the same results relative to the change of elongation, as 
the only effect of such motion will be to throw the points A, C, B 
forward in the orbit. It appears, then, that in the course of a sy- 
nodic revolution a superior planet will be seen at all angular dis- 
tances from the sun, both on the east and west side of him. From 
conjunction to opposition, that is, while the earth is passing from 
C to C, the planet will be to the right, or to the west of the sun ; 
and will therefore be below the horizon at sunset, and rise some 
time in the course of the night. But, from opposition to conjunc- 
tion, or while the earth is moving from C to C, it will be to the 
east of the sun, and therefore above the horizon at sunset. 

496. To find the length of the synodic revolution of a planet. — 
Let us first take an inferior planet, Venus for instance. Suppose 
we assume, at a given instant, the sun, Venus, and the earth to be 
in the same right line ; then, after any elapsed time, (a day for in- 
stance,) Venus will have described an angle m, and the earth an 
angle M around the sun. Now, m is greater than M ; therefore 
at the end of a day, the separation of Venus from the earth, (mea- 
suring the separation by an angle formed by two lines drawn from 
Venus and the earth to the sun,) will be m — M ; at the end of two 
days (the mean daily motions continuing the same) the angle of 
separation will be 2 (m — M) ; at the end of three days, 3 
(m — M) ; at the end of s days, s (m — M). When the angle of 
separation amounts to 360°, that is, when s(m — M) = 360°, the 
sun, Venus, and the earth must be again in the same right line, 
and in that case 

.-JSSL. . I. (128) . 

m — M v J 

In which expression s denotes the mean duration of a synodic 
revolution, m and M being taken to denote the mean daily motions. 

We may obtain from equation (128) another equation, in which 
the synodic revolution is expressed in terms of the sidereal periods 
of the earth and planet. 



SYNODIC REVOLUTIONS OF THE PLANETS. 187 

Let P and p denote the sidereal periods in question , then, since 

Id. : M° : : P : 360°, 

and 1 : m : : p : 360 ; 

M 360° , 360° , . . 

M = — pr-, and m = ; substituting, 

P p 



360°, 

p 



360° _ Pp 

\v P / 



(129). 



Equations (128), (129), although investigated for an inierioi 
planet, will answer equally well for a superior planet, provided wt 
regard m as standing for the mean daily motion of the earth, M foi 
that of the planet, p for the sidereal period of the earth, and P foi 
that of the planet. For the earth holds towards a superior planei 
the place of an inferior planet, and a synodic revolution of the earth 
to an observer on the planet, will obviously be a synodic revolu- 
tion of the planet to an observer on the earth. 

497. Equation (128) shows that the length of a mean synodic 
revolution depends altogether upon the amount of the difference 
of the mean daily motions of the earth and planet, and is the greater 
the less is this difference. 

It follows therefore that the synodic revolution is the longest for 
the planets nearest the earth. 

It appears by equation (129), that the length of a synodic revo- 
lution is, for an inferior planet, greater than the sidereal period of 
the planet, and for a superior planet, greater than the sidereal pe- 
riod of the earth. The actual lengths of the synodic .revolutions 
of the different planets are given in Table V. 

498. The mean synodic revolution of a planet being known, and 
also the time of one conjunction or opposition, we may easily as- 
certain its mean elongation at any given time, and thus approxi- 
mately the time of its rising, setting, and meridian passage. 

499. A planet will rise and set at the same hours at the end of a 
synodic revolution ; and will be an evening star, that is, above the 
horizon at sunset, during half of a synodic revolution, and a morn- 
ing star,«that is, above the horizon at sunrise, during an equal in- 
terval of time. The inferior planets will be evening stars from 
superior to inferior conjunction ; and the superior planets from op- 
position to conjunction. 

Mercury is an evening star for a period of 2 months ; Venus 
during an interval of 9^ months ; Mars for 1 year and 1 month ; 
Jupiter for Q\ months ; Saturn and Uranus each a few days more 
than 6 months. 

STATIONS AND RETROGRADATIONS OF THE PLANETS. 

500. The apparent motions of the planets in the heavens, as has 
already been stated (13), are not, like those of the suu and moon, 



188 



OF THE PLANETS AND THEIR PHENOMENA. 



continually from west to east, or direct, but are sometimes also 
from east to west, or retrograde. The retrograde motion takes 
place over arcs of but a small number of degrees ; and in changing 
the direction of their motions, the planets are for several days sta- 
tionary in the heavens. These phenomena are called the Stations 
and Retro gradations of the planets. We now propose to inquire 
theoretically into the particulars of the motions in question, and to 
show how the phenomena just mentioned result from the motions 
of the planets in connection with the motion of the earth. 

Let CAC'B (Fig. 84, p. 185) represent the orbit of an inferior 
planet, and PKT the orbit of the earth ; both considered as circles, 
and as situated in the same plane. If the earth were continually 
stationary in some point P of its orbit, it is plain that while the 
planet was moving from B the position of greatest western elonga- 
tion to A the position of greatest eastern elongation, it would ad- 
vance in the heavens from b to a ; that, while it was moving from 
A to B, that is, from greatest eastern to greatest western elonga- 
tion, it would retrograde in the heavens from a to b ; and that, in 
passing the points A and B, as it would be moving directly towards 
or from the earth, it would for a time appear stationary in the 
heavens in the positions a and b. 

But the earth is in fact in motion, and the actual apparent mo- 
tion of the planet is in consequence materially different from this. 
Let A, A' (Fig. 85) be the positions of the planet and earth at the 
time of the greatest eastern elongation, C, P their positions at in- 

Fig. 85. 




ferior conjunction, and B, B' their positions at the greatest western 
elongation. At the time of the greatest eastern elongation, while 



STATIONS AND RETROGRADATIONS OF THE PLANETS. 189 

the planet describes a certain distance AD on the line of the cen- 
tres' of the earth and planet, the earth moves forward in its orbit a 
certain distance A'D' ; so that, instead of appearing stationary at a 
in the interval, the planet will advance in the heavens from a to d. 
From the same cause it will have a direct motion about the time 
of the greatest western elongation. As it advances from A towards 
C, the direct motion will continue ; but, as the daily arc described 
by the planet will make a less and less angle with the daily arc de- 
scribed by the earth, the rate of motion will continually decrease, 
and finally, when the planet has come into a position with respect 
to the earth, such that the lines of direction of the planet, ??ip, m'p', 
at the beginning and end of the day are parallel, it will be station- 
ary in the heavens. As the daily arc of the planet is greater than 
that of the earth, and becomes parallel to it in inferior conjunction, 
the planet will be in the position in question before it comes into 
inferior conjunction. 

Subsequent to this, the inclination of the daily arcs still dimin- 
ishing, the lines of direction of the planet at the beginning and end 
of the day will diverge, and therefore the motion will be retro- 
grade. After inferior conjunction, the inclination of the arcs will, 
at corresponding positions of the earth and planet, obviously be the 
same as before. It follows, therefore, that the planet will be at its 
western station when it is at the same angular distance from the 
sun as at its eastern station ; that its motion will be retrograde un- 
til it has passed inferior conjunction and arrived at its western sta- 
tion ; and that after this it will be direct, q and n represent the 
positions of the planet and the earth at the time of the western sta- 
tion ; C'q = Cp, and Fn = Pm. 

The diminution of the elongation of the planet at its two stations 
is not the only effect of the earth's motion in the case under con- 
sideration; it also accelerates the direct, and retards the retrograde 
motion of the planet, and gives to the planet along with the sun an 
apparent motion of revolution around the earth. 

501. Let us now pass to the case of a superior planet. Sup- 
pose AC'B (Fig. 85) to be the orbit of the earth, and A'PB' that 
of the planet. Since the earth is an inferior planet to an observer 
stationed upon a superior planet, it appears by the foregoing arti- 
cle that it will, to an observer so situated, have a retrograde mo- 
tion while it is passing over a certain sacpC'q in the inferior part 
of its orbit, and a direct motion during the remainder of the sy- 
nodic revolution. Now, it is plain that the direction of the planet's 
motion, as seen from the earth, will always be the same as the di- 
rection of the earth's motion as seen from the planet. When the 
earth is at C, the middle of the arcpC'^, the planet is in opposi- 
tion. It follows, therefore, that a superior planet has a retrograde 
motion during a small portion of its synodic revolution, about the 
lime of opposition. (See Table V.) 



190 



OF THE PLANETS AND THEIR PHENOMENA. 



PHASES OF THE INFERIOR PLANETS. 



502. To the naked sight the disc of the planet Venus appears 
circular, like that of each of the other planets, but the telescope 
shows this to be an optical illusion. When Venus is repeatedly 
observed with a telescope, it is seen to present in its various posi- 
tions with respect to the sun the same variety of phases as the 
moon ; being a full circle at superior conjunction, a half circle at 
the greatest eastern and western elongations, and a crescent, with 
the horns turned from the sun, before and after inferior conjunction. 

Mercury exhibits precisely similar phases, but being smaller, at 
a greater distance from the earth, and much nearer the sun, its 
phases are not so easily observed as those of Venus. 

503. The phases of Venus are easily accounted for, by suppos- 
ing it to be an opake spherical body, and to shine by reflecting the 
sun's light, and by taking into consideration its motion with respect 

to the sun and earth. The hemi- 
sphere turned towards the sun is 
illuminated by him, and the other is* 
in the dark, and as the planet re- 
volves around the sun, various por- 
tions of the enlightened half are 
turned towards the earth : in supe- 
rior conjunction, the whole of it ; at 
the greatest elongations, one half; 
and near inferior conjunction, but a 
small part. This will be abundant- 
ly evident on inspecting Fig. 86. 
The phases corresponding to the 
positions represented are delineated 
in the figure. 

The phases of Mercury are ob- 
viously susceptible of a similar ex- 
planation. 

504. The disc of the planet Mars also undergoes changes of 
form, but they are of comparatively moderate extent. It is some- 
times gibbous, but never has the form of a crescent. Indeed, on 
the supposition that Mars is an opake body illuminated by the sun, 
we would not see the whole of the enlightened hemisphere, except 
in conjunction and opposition, but there would always be more 
than half of it turned towards the earth, and therefore the disc 
should always be larger than a half circle. 

505. The discs of the other superior planets do not experience 
any perceptible variation of form, for the reason, doubtless, that 
their orbits are so large with respect to the orbit of the earth, that 
all, or very nearly all of their illuminated hemispheres, is con- 
stantly visible from the earth. 




TRANSITS OF THE INFERIOR PLANETS. 191 



TRANSITS OF THE INFERIOR PLANETS. 

506 The two inferior planets Venus and Mercury, at inferior 
conjunction, sometimes, though rarely, pass between the sun and 
earth, and are seen as a dark spot crossing the sun's disc. This 
phenomenon is called a Transit. It will take place, in the case 
of either planet, whenever, at the time of inferior conjunction, it is 
so near either node that its geocentric latitude is less than the ap- 
parent semi-diameter of the sun. 

507. The transits of Venus take place alternately at intervals of 
8 and 105J- or 12H years. The last were in the years 1761 and 
1769. The next will be in 1874 and 1882 ; of which the latter 
will be visible in this country. 

In consequence of the greater distance of Mercury from the 
earth, a greater portion of its orbit is directly interposed between 
the sun and earth than of the orbit of Venus ; moreover, the sy- 
nodic revolution of Mercury is shorter than that of Venus. On 
these accounts, it happens that the transits of Mercury are much 
more frequent than those of Venus. The last transit of Mercury 
was on May 8th, 1845. The next two will take place in 1848, 
and 1861, in the month of November. The first, which will oc- 
cur on the 9th, will be visible in this country. 

508. A transit is calculated in a precisely similar manner with a 
solar eclipse ; the planet in the one calculation answering to the 
moon in the other. 

509. A transit is an important phenomenon in a practical point 
of view, as it furnishes the most exact means we possess of ascer- 
taining the sun's parallax. In order to understand how this phe- 
nomenon can be used for this purpose, we have only to consider 
that, in consequence of the difference of the parallaxes of the sun 
and Venus, observers at different stations upon the earth will refer 
the planet to different points upon the sun's disc, and that therefore, 
to such observers, the transit will take place along different chords, 
and be accomplished in unequal portions of time. This fact is rep- 
resented to the eye in Fig. 87. It is then to be expected, that, if the 
durations of the transit at two different places should be noted, the 

Fig. 87. 




difference of the parallaxes of the sun and Venus, upon which 
alone the difference of the duration depends, could be computed 
This computation is in fact possible. Also, the ratio of the paral 
lixes being inversely as that of the distances, could be found by the 



192 OF THE PLANETS AND THEIR PHENOMENA. 

elliptical theory of the planetary motions, and thus the parallax both 
of the sun and Venus would become known. 

510. The parallax of the sun, as it is now known, was deduced 
from observations upon the transits of Venus in 1769 and 1761. 
Expeditions were fitted out on the most efficient scale, by the 
British, French, Russian, and other governments, and sent to va- 
rious parts of the earth, remote from each other, to observe the 
transit of 1769, that the parallax of the sun might be computed 
from the results of the observations. The sun's parallax, as de- 
termined by Professor Encke from the observations made upon the 
transit in question, and that of 1761, is 8 // .5776. 

APPEARANCES, DIMENSIONS, ROTATION, AND PHYSICAL 
CONSTITUTION OF THE PLANETS. 

511. It appears from admeasurement with the telescope and 
micrometer, that the apparent diameter of a planet is subject to 
sensible variations. The apparent diameter of Venus, as well as 
of Mercury, is greatest in inferior conjunction, and least in superior 
conjunction ; while the apparent diameter of each of the other 
planets is greatest in opposition and least in conjunction. These 
variations of the apparent diameters of the planets, are necessary 
consequences of the changes that take place in the distances of 
the planets from the garth. (See Fig. 84.) 

512. The real diameter of a planet is deduced from its apparent 
diameter and horizontal parallax. (See Art. 429.) When the di- 
ameters of the planets have been found, their relative surfaces and 
volumes are easily obtained ; for the surfaces are as the squares 
of the diameters, and the volumes as the cubes. 

513. The order of magnitude of the planets is as follows: 
1 Jupiter, 2 Saturn, 3 Uranus, 4 the Earth, 5 Venus, 6 Mars, 
7 Mercury, 8 Pallas. 9 Ceres, 10 Juno, 11 Vesta. The range of 
magnitude, for the principal planets, is from 1 to about 20,000. 
(The relative magnitudes of the planets are represented to the eye 
in the Frontispiece.) (See Note VIII.) 

514. Spots more or less dark have been seen upon the discs of 
most of the principal planets ; and by passing across them from 
east to west and reappearing at the eastern limbs, have established 
that the planets upon which they are observed rotate upon axes 
from west to east. * From repeated careful observations upon the 
situations of these spots, the periods of rotation, and the positions 
of the axes, have been determined. (See Note IX.) 

The periods of rotation of Mercury, Venus, the Earth, and Mars, 
are all about 24 hours, and of Jupiter and Saturn about 10 hours. 
Those of the other planets are not known. The axes of rotation 
remain continually parallel to themselves, as the planets revolve in 
their orbits. 

515. The amount of light ana\heat, which the sun bestows upon 



MERCURY VENUS. 193 

the planets, decreases as we recede from the sun, in the same ratio 
that the square of the distance increases. (See Table IV.) 

516. It will be seen in the sequel that the planets are all opake 
bodies, like the earth ; and that they are surrounded with an atmo- 
sphere, after the same manner as the earth. 

MERCURY. 

517. In consequence of its proximity to the sun, Mercury is 
rarely visible to the naked eye. When seen under the most favor- 
able circumstances about the time of greatest elongation, it presents 
the appearance of a star of the 3d or 4th magnitude. Its phases 
show that it is opake, and illuminated by the sun. Its apparent 
diameter varies with its distance from 5" to 12". Its real diame- 
ter is about 3000 miles, or § of that of the earth, and its volume 
is about T V of the earth's volume.* 

Mercury performs a rotation on its axis in 24h. 5Jm., and its 
axis is inclined to the ecliptic under a small angle. 

518. Owing to the dazzling splendor of its rays, and the tremulous motion in- 
duced by the ever-varying density of the air and vapors near the earth's surface, 
through which it is seen, the telescope does not present a well-defined image of the 
disc of this planet. Schroeter is the only observer who has ever detected any spots 
upon it. From the fact that spots are only occasionally seen, it has been inferred 
that the planet is surrounded with a dense atmosphere, which reflects a strong light, 
and, except when it is particularly pure, prevents the darker body of the planet 
from being seen. 

Schroeter, in making observations upon Mercury at the time his disc had the 
form of a crescent, discovered that one of the horns of the crescent became blunt 
at the end of every 24 hours : from which he inferred that the planet turned upon 
an axis, and had mountains upon its surface, which were brought at the end of 
every rotation into the same position with respect to his eye and the sun. 

VENUS. 

519. Venus is the most brilliant of all the planets, and generally 
appears larger and brighter than any of the fixed stars. At times, 
it emits so much light as to be visible at noonday. It is found by 
calculation, that the epochs in the course of a synodic revolution, 
at which Venus gives most light to the earth, are those at which, 
bein"" in the inferior part of its orbit, it has an elongation of about 
40°. They are about 36 days before and after inferior conjunc- 
tion. The disc is then considerably less than a semicircle, but the 
increased proximity to the earth more than compensates for the 
diminished size of the disc. Venus will besides attain to greater 
splendor in some revolutions than others, in consequence of being 
nearer the earth, when in the most favorable position. 

520. As seen through a telescope, Venus presents a disc of 
nearly uniform brightness, and spots have very rarely been seen 
upon it. Its phases prove it to be an opake spherical body, 
shining by reflecting the sun's light. Its apparent diameter varies 
with its distance from 10" to 61". Its real diameter is about 7800 

* The exact diameters, volumes, times of rotation, &c, of the different planets, 
as far as known, may be found in Table I~ T . 

25 



194 OF THE PLANETS AND THEIR PHENOMENA. 

miles, and its volume about Jjless than that of the earth. The 
period of its rotation is 23h. 21m. The inclination of its axis to 
the plane of its orbit is not exactly known, but is not far from 18°. 

521. From the remarkable vivacity of the light of this planet, which far ex- 
ceeds that of the light reflected from the moon's surface, as well as the transitory 
nature of the few darkish spots which have been seen upon its disc, it is inferred 
that it is surrounded by a dense and highly reflective atmosphere, which in gene- 
ral screens the whole of the darker body of the planet from our view. The truth 

Fig. 88. ' °f tn i s inference is confirmed by certain deli- 

cate observations made by Schroeter. This 
astronomer distinctly discerned a faint bluish 
light stretching beyond the proper termination 
of one of the horns of the crescent into the dark 
part of the face of the planet, as is represented 
in Fig. 88, where the left extremity of the dot- 
ted line represents the natural terminating point 
of one of the horns of the crescent. This he 
considered to be a twilight on the surface of 
Venus. 

Since the transparency of Venus's atmosphere 
is variable, becoming occasionally such as to 
admit of the body of the planet's being seen 
through it, we must suppose that it contains 
aqueous vapor and clouds, and therefore that 
there are bodies of water upon the surface of the planet. It is in fact supposed 
that isolated clouds have actually been seen. The most natural explanation of 
the bright spots which have sometimes been noticed on the disc is, that they are 
clouds more highly reflective than the atmosphere or than the clouds in general. 

522. There are great inequalities on the surface of Venus, and, it would seem, 
mountains much higher than any upon our globe. Schroeter detected these masses 
by several infallible marks. In the first place the edge of the enlightened part of 
Venus is shaded, as seen in Figs. 88, 89, and 90, and as the moon appears when 
in crescent even to the naked eye. This appearance is doubtless caused by shad- 
ows cast by mountains ; which are naturally best seen on that part of the planet 
to which the sun is rising or setting, where they are longest. In the next place, 
the edge of the disc shows marked irregularities. Thus it often appears rounded 
at the corners, as in Fig. 89, owing undoubtedly to part of the disc being rendered 
invisible there by the shadow or interposition of some line of eminences ; and at 

Fig. 89. Fig. 90. 






other times, as in Fig. 90, a single bright point appears detached from the disc — 
the top of a high mountain, illuminated across a dark valley. 

Schroeter found that these appearances recurred regularly at equal intervals of 
about 23 £ hours ; the same period as that which Cassini had previously found for 
the completion of a rotation, by observations upon the spots. 



MARS. 195 



MARS. 



523 Mars is of the apparent size of a star of the first or second 
magnitude, and is distinguished from the other planets by its red 
and fiery appearance. The observed variation in the form of its 
disc (504) shows that it derives its light from the sun. Its greatest 
and least apparent diameters are respectively 4" and 18", Its real 
diameter is something over 4000 miles, or rather more than \ of 
the diameter of the earth, and its bulk is about { of that of the 
earth. 

Mars revolves on its axis in 24h. 37m. ; and its axis is inclined 
to the ecliptic in an angle of about 60°. It appears, from meas- 
urements made with the micrometer, that its polar diameter is less 
than the equatorial, and thus, that, like the earth, it is flattened at 
its poles. According to Sir W. Herschel, its oblatehess (159) 
is T V: according to Arago ^. 

524. When the disc of Mars is examined with telescopes of 
great power it is generally seen to be diversified with spots of dif- 
ferent shades, which, with occasional variations, retain constantly 
the same size and form. 

They are conjectured to be continents and seas. In fact, Sir J. F. W. Her- 
schel has on several occasions, in examining this planet with a good telescope, no- 
ticed that some of its spots are of a reddish color, while others have a greenish 
•dnge. The former he supposes to be land, and the latter water. Fig. 91 repre- 
sents Mars in its gibbous state as YW. 91. 
seen by Herschel in his 20 feet re- 
flector, on the 16th of August, 
1830. The darker parts are seas. 
The bright spot at the top is at 
one of the poles of Mars. At 
•other times a similar bright spot is 
seen at the other pole. These 
brilliant white spots have been 
■conjectured with a great deal of 
probability to be snow ; as they 
are reduced in size, and sometimes 
disappear when they have been 
long exposed to the sun, and are 
greatest when just emerging from 
the long night of their polar winter, 

525. The great divisions of the 
surface of Mars are seen with dif- 
ferent degrees of distinctness at 

different times, and sometimes disappear, either partially or entirely : parts oi uus 
disc also appear at times particularly dark or bright. From these facts it is to be 
inferred that this planet is environed with an atmosphere, and that this contains 
aqueous vapor which, by varying in quantity and density, renders its transpa- 
rency variable. 

526. No mountains have been detected upon Mars. But this is no good reason 
for supposing that they are really wanting there ; for, if the surface of Mars be 
actually diversified with mountains and valleys, since its disc never differs much 
from a full circle, we have no reason to expect that its edge would present that 
shaded appearance and those irregularities which have been noticed on Venus and 
Mercury, when of the form of a crescent. The same remarks will apply with still 
greater force to the other superior planets. 

527. The ruddy color of the light of Mars has generally been attributed to its 




196 OF THE PLANETS aXD THEili PHENOMENA. 

atmosphere, but Sir John Herschel finds a sufficient cause for this phenomenon ii» 
the ochrey tinge of the general soil of the planet (534.] 

JUPITER AND ITS SATELLITES. 

528, Jupiter is the most brilliant of the planets, except Tenus y 
and sometimes even surpasses Venus in brightness. The eclipses 
of its satellites prove that it is an opake body, and that it shines 
by reflecting the light of the sun. Its apparent diameter, when 
greatest, is 46", and when least, 30". 

Jupiter is the largest of all the planets. Its diameter is about 
11 times the diameter of the earth, or about 87, 000 miles, and its 
bulk is more than 1200 times that of the earth. It turns on an 
axis nearly perpendicular to the ecliptic, and completes a rotation 
in 9h. 56m . The polar diameter is about T ] T less than the equa- 
torial. 

529. When Jupiter is examined with a good telescope, its disc 
^s always observed to be crossed by several obscure spaces, which 
are nearly parallel to each other, and to the plane of the equator. 

These are called the Belts of 
Jupiter. (See Fig. 92, which 
mts the appearance of 
Jupiter as seen by Sir John 
Herschel in his twenty-feet 
reflector, on the 23d of Sep- 
tember, 1832.) They vary 
somewhat in number, breadth, 
and situation on the disc, but 
never in direction. Sometimes 
only one or two are visible ; on 
other occasions as many as 
eight have been seen at the 
same time. Sir William Her- 
schel even saw them on one or 
occasions broken up and distributed over the whole face of the 
planet : but this phenomenon is extremely rare. Branches run- 
ning out from the behs and subdivisions, as represented in the 
figure, are by no means uncommon. Dark spots of invariable 
form and size have also been seen upon them. These have been 
observed to have a rapid motion across the disc, and to return at 
equal intervals to the same position on the disc, after the same 
manner as the sun* s spots ; which leaves no room to doubt that they 
are on the body of the planet, and that this turns upon an axis. 
Bright spots have also been noticed upon the belts. The belts 
generally retain pretty nearly the same appearance for several 
months together, but occasionally marked changes of form and 
size have taken place in the course of an hour or two. 

The occasional variations of Jupiter's belts, and the occurrence of spots upon 
them, which are undoubtedly permanent portions of the mass of the planet, render 
it extremely probable that they are the body of the planet seen through an atmc*- 




JUPITER SATURN. 197 

sphere of variable transparency ; but in general having extensive tracts of compar- 
atively clear sky in a direction parallel to the equator. These are supposed to be 
determined by currents analogous to our trade winds, but of a much more steady 
and decided character ; as would be the necessary consequence of the superior 
velocity of rotation of this planet. As remarked by lierschei, that it is the com- 
paratively darker body of the planet which appears in the belts, is evident from 
this, — that they do not come up in all their strength to the edge of the disc, but 
fade away gradually before they reach it. 

The bright belts, intermediate between the dark ones, are probably bands of 
clouds or tracts of less pure air. 

530. The satellites of Jupiter, as it has been already remarked, 
are visible with telescopes of very moderate power. With the 
exception of the second, which is a little smaller, they are some- 
what larger than the moon. The orbits of the satellites lie very 
nearly in the plane of Jupiter's equator. They are therefore all 
viewed nearly edgewise from the earth, and in consequence the 
satellites always appear nearly in a line with each other. 

531. Sir W. Herschel, in examining the satellites of Jupiter 
with a telescope, perceived that they underwent periodical varia- 
tions of brightness. These variations he supposed to proceed from 
a rotation of the satellites upon axes, which caused them to turn 
different faces towards the earth ; and from repeated and careful 
observations made upon them, he discovered that each satellite 
made one turn upon its axis in the same time that it accomplished 
a revolution around the primary; and therefore, like the moon, ' 
presented continually the same face to the primary. ! 

SATURN, WITH ITS SATELLITES AND RING. 

532. Saturn shines with a pale dull light Its apparent diame- 
ter varies only 3" or 4" by reason of the change of distance, and 
Is at the mean distance about 16". The eclipses of its satellites 
prove that it is opake and illuminated by the sun. 

Saturn is the largest of the planets, next to Jupiter. Its diame- 
ter is about 10 times the diameter of the earth, or 79,000 miles ; 
and its volume is about 900 times that of the earth. The rotation 
on its axis is performed in lOh. 29m. The inclination of its axis 
to the ecliptic is about 60°. Its oblateness is T V 

533. The disc of Saturn, like that of Jupiter, is frequently 
crossed with dark bands' or belts, in a direction parallel to its equa- 
tor. Extensive dusky spots are also occasionally seen upon its 
surface. (See Fig. 93.) 

The cause of Saturn's belts is doubtless the same as of Jupiter's. They accord- 
ingly prove the existence of an atmosphere and of aqueous vapor, and thus also of 
bodies of water, upon the surface of Saturn. 

534. The planet Saturn is distinguished from all the other 
planets in being surrounded by a broad, thin, luminous ring, situ- 
ated in the plane of its equator, and entirely detached from the 
body of the planet. (See Fig. 93.) This ring sometimes casts a 
shadow upon the planet, and is, in turn, at times partially obscured 



198 



OF THE PLANETS ANI> THEIR PHENOMENA. 




by the shadow of the planet ; from which we conclude that it is 
opake, and receives its light from the sun- 
Fig. 93. It is inclined to the 
plane of the ecliptic in an 
angle of about 28°, and 
during the motion of Sat- 
urn in its orbit it remains 
continually parallel to it- 
self. The face of the ring 
is, therefore, never viewed 
perpendicularly from the 
earth, and for this reason 
never appears circular, al- 
though such is its actual 
form. Its apparent form 
is that of an ellipse, more 
or less eccentric, accord- 
ing to the obliquity under which it; is viewed, which varies with 
the position of Saturn in its orbit. When it is seen under the 
larger angles of obliquity, it appears as a luminous band nearly 
encircling the planet, mid is visible in telescopes of small power. 
Stars can also be seen between it and the planet in these positions. 
At other times, when viewed very obliquely, it can be seen only 
with telescopes of high power. When it is approaching the latter 
state, it has the appearance of two handles or ansce, one on each 
side of the planet. 

It is also at times invisible. This is the case whenever the 
earth and sun are on different sides of the plane of the ring, for 
the reason that the illuminated face is then turned from the earth. 
When the plane of the ring passes through the centre of the sun, 
the illuminated edge can be seen only in telescopes of extraordi- 
nary power, and appears as a thread of light cutting the disc of 
the planet 

535. Since the orbit of Saturn is very large in comparison with 
the orbit of the earth, the plane of the ring, during the greater 
part of the revolution of Saturn, will pass without the orbit of the 
earth ; and when this is the case the ring will be visible, as the 
earth and sun will be on the same side of its plane. During the 
period, which is about a year, that the plane of the ring is passing 
by the orbit of the earth, the earth will sometimes be on the same 
side of it as the sun, and sometimes on opposite sides. In the 
latter case the ring will be invisible, and in the former will be seen 
so obliquely as to be visible only in telescopes of considerable or 
great power. All this will perhaps be better understood on con- 
sulting Fig. 94, where efg represents the orbit of the earth. The 
appearances of the ring in the different positions of the planet in its 
orbit are delineated in the figure 

The plane of the ring will pass through the sun every semi- 



SATURN S RING. 



199 



revolution of Saturn, or, at a mean, about every 15 years, and at 
the epochs at which the longitude of the planet is respectively 
170° and 350°. The ring will then disappear once in about 15 
years ; but, owing to the different situations of the earth in its or- 

Fig. 94. 




bit, under circumstances oftentimes quite different. And the dis- 
appearance will occur when the longitude of the planet is about 
170°, or 350°. The ring will be seen to the greatest advantage 
when the longitude of the planet is not far from 80° or 260°. 
The last disappearance took place in 1833 ; the next will be in 1847. 
At the present time (1845) the north face of the ring is visible. 

536. From observations made upon bright spots seen on the 
face of the ring, Herschel discovered that it revolved from west to 
east about an axis perpendicular to its plane, and passing through 
the centre of the planet, (or very nearly.) The period of its ro- 
tation is lOh. 32m. It is remarkable that this is the period in 
which a satellite assumed to be at a mean distance equal to the 
mean distance of the particles of the ring, would revolve around 
the primary according to the third law of Kepler. 

The breadth of the ring is about one-half greater than its dis- 
tance from the surface of the planet, and is about equal to one- 
third the diameter of the planet, or 29,000 miles. 

537. What we have called Saturn's ring consists in fact of two 
concentric rings, which turn together, although entirely detached 
from each other. The void space between them is perceived in 
telescopes of high power, under the form of a black oval line. 
According to the calculations of Sir John Herschel, from the mi- 
crometric measures of Professor Struve, the breadth of the interior 
ring is about 17,200 miles, and of the exterior about 10,600 miles; 
the interval between the rings is nearly 1800 miles, and the dis- 
tance from the planet to the inside of the interior ring is a little 
over 19,000 miles. The thickness of the rings is not well known; 
the edge subtends an angle much less than 1", which, at the dis- 
tance of the planet, answers to about 5000 miles. Herschel makes 
it less than 250 miles. (See Note X.) 

538. Professor Bessel has shown that the double ring is not bounded by parallel 
plane surfaces. He infers this to be the case from the fact that at almost every 



200 OF THE PLANETS AND THEIR PHENOMENA. 

disappearance or reappearance of the ring, the two ansae have not disappeared or 
reappeared at the same time. He has also found, from a discussion of the obser- 
vations which have been made upon the disappearances and reappearances of the 
ring', that they cannot be satisfied by supposing the two faces of the ring to be 
parallel planes. In view of all the facts, it seems most probable that the cross sec- 
tion of each ring is a very eccentric ellipse, instead of a rectangle, and that it varies 
somewhat in size from one part of the ring to another. It may have irregularities 
on its surface as great or greater than those which diversify the surface of the earth. 

539. Whatever may be the form of the rings, their matter is not uniformly dis- 
tributed. For recent micrometric measurements of great delicacy, made by Pro- 
fessor Struve, have made known the fact, that the rings are not concentric with 
the planet, but that their centre of gravity revolves in a minute orbit about the 
centre of the planet. Laplace had previously inferred, from the principle of gravi- 
tation, that this circumstance was essential to the stability of the rings. He de- 
monstrated that if the centre of gravity of either ring were once strictly coincident 
with the centre of gravity of the planet, the slightest disturbing force, such as the 
attraction of a satellite, would destroy the equilibrium of the ring, and eventually 
cause the ring to precipitate itself upon the planet. 

540. In respect to the origin of Saturn's ring, Sir John Herschel has offered the 
interesting suggestion, that, as the smallest difference of velocity in space between 
the planet and ring must infallibly precipitate the latter on the former, never more 
to separate, it follows either that their motions in their common orbit around the 
sun must have been adjusted by an external power with the minutest precision, 
or that the ring must have been formed about the planet while subject to their 
common orbitual motion, and under the full and free influence of all the acting 
forces. The tatter supposition accords with Laplace's theory of the progressive 
creation of the universe, hereafter to be noticed. 

541. The satellites of Saturn were discovered, the 6th in the 
order of distance by Huygens, in 1655, with a telescope of 12 feet 
focus ; the 3d, 4th, 5th, and 8th, by Dominique Cassini, between 
the years 1670 and 16S5, with refracting telescopes of 100 and 
136 feet in length ; and the 1st and 2d by Sir William Herschel, 
in 1789, with his great reflecting telescope of 40 feet focus. All 
but the 1st and 2d are visible in a telescope of a large aperture, 
with a magnifying power of 200. (See Note XL) 

They all, with the exception of the 8th, revolve very nearly in 
the plane of the ring and of the equator of the primary. The or- 
bit of the 8th is inclined under a considerable angle to this plane, 
According to Sir John Herschel, the 6th satellite is much the lar- 
gest, and is estimated to be not much inferior to Mars in size. The 
others diminish in size as we proceed inward ; until the 1st and 2d 
are so small, and so near the ring, that they have never been dis- 
cerned but with the most powerful telescopes which have yet been 
constructed ; and with these only at the time of the disappearance 
of the ring, (to ordinary telescopes,) when they have been seen as 
minute points of light skirting the narrow line of the luminous 
edge of the ring. 

The 8th satellite is subject to periodical variations of lustre, 
which prove its rotation on an axis in the period of a sidereal revo- 
lution of Saturn. 

URANUS AND ITS SATELLITES. 

542. Uranus is scarcely visible to the naked eye. In a tele- 
scope it appears as a small round uniformly illuminated disc. Its 



URANUS VESTA JUNO — CERES — PALLAS. 201 

apparent diameter is about 4", from which it never varies much, 
owing to the smallness of the earth's orbit in comparison with its 
own. Its real diameter is about 34,500 miles, and its bulk 82 
times that of the earth. Analogy leads us to believe that this pla- 
net is opake and turns on an axis, but there is no direct proof that 
this is the case. 

543. The satellites of Uranus were discovered by Sir W. Her- 
schel. They are discernible only with telescopes of the highest 
power. (See Note XII.) 

VESTA JUNO CERES PALLAS. 

544. These four planets, although less distant than several of 
the others, are so extremely small, that they cannot be seen with- 
out the aid of a telescope. 

Vesta is the most brilliant, and shines with a white light. In 
the telescope it appears as a star of about the 6th magnitude. Juno 
and Ceres have the apparent size of a star of the 8th magnitude ; 
and together with Pallas have a ruddy aspect and a variable lus- 
tre, indicative of the presence of atmospheres of variable density 
and purity. Ceres and Pallas generally shine with a pale dull 
light, and are seen surrounded with a nebulosity, or haziness of, 
according to Herschel, from three to six times the extent of the 
body of the planet. This haziness is sometimes so decided as to 
conceal the body of the planet from view, and at other times en- 
tirely disappears, leaving the disc of the planet sharply denned and 
alone visible. 

545. The actual magnitudes of these planets are not well known. 
The determinations of different Astronomers are widely different. 
The following are perhaps the nearest approximations to their true 
diameters that have yet been obtained : Vesta 270 miles ; Juno 
460 miles ; Ceres 460 miles ; Pallas 670 miles. 



CHAPTER XVII. 



OF COMETS. 
THEIR GENERAL APPEARANCE— VARIETIES OF APPEARANCE. 



* 



546. The general appearance of comets is that of a mass of 
some luminous nebulous substance, to which the name Coma has 
been given, condensed towards its centre around a brilliant Xucleus 
that is in general not very distinctly defined, from which proceeds 
m a direction opposite to the sun a fainter stream or train of simi- 
lar nebulous matter, called the Tail. The coma and nucleus to- 
gether form what is called the Head of the Comet. (See Fig. 95.) 

26 



202 OF COMETS. 

The tail gradually increases in width, and at the same time di- 
minishes in distinctness from the head to its extremity, where it is 
generally many times wider than at the head, and fades away un- 

Fig. 95. 




Great Comet of 1811. 

til it is lost in the general light of the sky. It is, in general, less 
bright along its middle than at the borders. From this cause the tail 
sometimes seems to be divided, along a greater or less portion of 
its length, into two separate tails or streams of light, with a com- 
parative dark space between them. Ordinarily it is not straight, 
that is, coincident with a great circle of the heavens, but concave 
towards that part of the heavens which the comet has just left. 
This curvature of the tail is most observable near its extremity. 
The most remarkable example is that of the comet of 1 744, which 
was bent so as to form nearly a quarter of a circle. Nor does the 
general direction of the tail usually coincide exactly with the great 
circle passing through the sun and the head of the comet, but de- 
viates more or less from this, the position of exact opposition to the 
sun in the heavens, on the side towards the quarter of the heavens 
just traversed by the comet. This deviation is quite different for 
different comets, and varies materially for the same comet while it 
continues visible. It has even amounted in some instances to a 
right angle. 

547. The apparent length of the tail varies from one comet to 
another from zero to 100° and more ; and ordinarily the tail of the 
same comet increases and diminishes very much in length during 



GENERAL APPEARANCE OF COMETS. 



203 



the period of its visibility. When a comet first appears, in general, 
no tail is perceptible, and its light is very faint. As it approaches 
the sun, it becomes brighter : the tail also after a time shoots out 
from the coma, and increases from day to day in extent and dis- 
tinctness. As the comet recedes from the sun, the tail precedes 
the head, being still on the opposite side from the sun, and grows 
less and less at the same time that, along with the head, it de- 
creases in brightness, till at length the comet resumes nearly its 
first appearance, and finally disappears. (See Fig. 97.) It some- 
times happens that, owing to peculiar circumstances, Fig. 96. 
a comet does not make its appearance in the firm a- MjWjWHiH 
ment until after it has passed the sun in the heavens, m 
and not until it has attained to more or less distinct- B 
ness, and is furnished with a tail of considerable or m 
even great length. This was remarkably the case $§| 
with the great comet of 1843. (See Art. 326 ; also S 
Fig. 96.) 

548. The tail of a comet is the longest, and the p 
whole comet is intrinsically the most luminous, not g 
long after it has passed its perihelion. Its apparent H 
size and lustre will not, however, necessarily be the m 
greatest at this time, as they will depend upon the ^^Rfe^^^B 
distance and position of the earth, as well as the ac- HR: 
tual size and intrinsic brightness of the comet. To I 

Fi g- 97 - wmiimH 




illustrate this, let abed (Fig. 97) represent the orbit j 
of the earth, and MPN the orbit of a comet, having 
its perihelion at P. Now, if the earth should chance 
to be at a when the comet, moving towards its peri- 
helion, is at r, it might very well happen that the 
comet would appear larger and more distinct than | 



204 OF COMETS. 

when it had reached the more remote point s, although when at the 
latter point it would in reality be larger and brighter than when at 
r. It would be the most conspicuous possible if the earth should 
be in the vicinity of c or b soon after the perihelion passage : and 
it would be the least conspicuous possible if the comet, sup- 
posed to be moving in the direction XPM, should pass from N 
around to M, while the earth is moving aroimd from a to b or c, so 
as to be continually comparatively remote from the comet, and sc 
that the comet will be in conjunction with the sun at the time after 
the perihelion passage when its actual size and intrinsic lustre are 
the greatest. It is to be observed that the apparent lustre of a 
comet is sometimes very much enhanced by the great obliquity of 
the tail, in some of its positions, to the line of sight. This seems 
to have been the case with the comet of 1543, on February 25th, 
(see Fig. 56,) and was doubtless one reason of its being so very 
bright as to be seen in open day in the immediate vicinity of the 
sun. 

Since the earth may have every variety of position in its orbit 
at the different returns of the same comet to its perihelion, it will 
be seen, on examining Fig. 97, that the circumstances of the ap- 
pearance and disappearance of the comet, as well as its size and 
distinctness, may be very various at its different returns. This 
has been strikingly true in the case of Hallev's Comet. Gamb art's 
Comet was also invisible in its return to its perihelion in 1S39, by 
reason of its continual proximity to the line of direction of the sun 
as seen from the earth, and its great distance from the earth. 

549. Individual comets offer considerable varieties of aspect. 
Some comets have been seen which were wholly destitute of a 
tail : such, among others, was the comet of 1652, which Cassini 
describes as being as round and as bright as Jupiter. Others have 
had more than one luminous train. The comet of 1744 was pro- 
vided with six, which were spread out, like an immense fan, 
through an angle of 117° ; and that of 1523 with two, one directed 
from the sun in the heavens, and, what is very remarkable, another 
smaller and fainter one directed towards the sun. Others still have 
had no perceptible nucleus, as the comets of 1795 and 1504. 

The comets that are visible only in telescopes, which are very 
numerous, have, generally, no distinct nucleus, and are often entire- 
ly destitute of every vestige of a tail. They have the appearance of 
round masses of luminous vapor, somewhat more dense towards the 
centre. Such are Encke's and Biela's comets. (See Fig. 95.) The 
point of greatest condensation is often more or less removed from the 
centre of figure on the side towards the sun ; and sometimes also 
on the opposite side. (See Xote XIII.V 

550. The cornets which have had the longest tails are those of 
1650, 1769, and 1615. The tail of the greaf comet of 1650, when 
apparently the longest, extended to a distance of 70° from the head ; 
that of the comet of 1769, a distance of 97 ° ; and that of the com- 



FORM, STRUCTURE, AND DIMENSIONS OF COMETS. 205 

et of 1618, 104°. These are the apparent lengths as seen at cer- 
tain places. By reason of the different degrees of purity and den- 
sity of the air through which it is seen, the tail of the same comet 
often appears of a very different length to observers at different 

Fig. 98. 




Enckc's Comet. 

places. Thus, the comet of 1769, which at the Isle of Bourbon 
seemed to have a tail of 97° in length, at Paris was seen with a tail 
of only 60°. From this general fact we may infer that the actual 
tail extends an unknown distance beyond the extremity of the ap- 
parent tail. 

FORM, STRUCTURE, AND DIMENSIONS OF COMETS. 

551. The general form and structure of comets, so far as they 
can be ascertained from the study of the details of their appear- 
ance, may be described as follows : The head of a comet consists 
of a central nucleus, or mass of matter brighter and denser than 
the other portions of the comet, enveloped on the side towards 
the sun, and ordinarily at a great distance from its surface in 
comparison with its own dimensions, by a globular nebulous 
mass of great thickness, called the Nebulosity, or nebulous En- 
velope. This, it is said, never completely surrounds the nu- 
cleus, except in the case of comets which have no tails. It forms 
a sort of hemispherical cap to the nucleus on the side towards the 
sun. Its form, however, is not truly spherical, but approximates 
to that of an hyperboloid having the nucleus in its focus and its ver- 
tex turned towards the sun. The tail begins where the nebulosity 
terminates, and seems, in general, to be merely the continuation of 
this in nearly a straight line beyond the nucleus. There is ordina- 
rily, as has been already intimated, a distinct space containing but 
little luminous matter between the nucleus and the nebulosity, but 
this is not always the case. The tail of a comet has the shape of 
a hollow truncated cone, with its smaller base in the nebulosity of 
the head j with this difference, however, that the sides are usually 



206 OF COMETS. 

more or less curved, and ordinarily concave towards the axis. That 
the tail is hollow is evident from the fact, already noticed, that on 
whichever side it is viewed it appears less bright along the middle 
than at the borders. There can be less luminous matter on a line 
of sight passing through the middle, than on one passing near one 
of the edges, only on the supposition that the tail is hollow. The 
whole tail is generally bent so as to be concave towards the regions 
of space which the comet has just left. 

552. In some instances the nucleus is furnished with several 
envelopes concentric with it : which are formed in succession as 
the comet approaches the sun. For example, the comet of 1744, 
eight days after the perihelion passage, had three envelopes. Some- 
times each of them is provided with a tail. Each of these sev- 
eral tails lying one within the other, being hollow, may in conse- 
quence appear so faint along its middle as to have the aspect of 
two distinct tails. A comet which has in reality three separate tails, 
might thus appear to be supplied with six, as was the comet of 
1744. If the different envelopes were not distinctly separate from 
each other, then we should have all the tails appearing to proceed 
from the same nebulous mass. 

553. Supernumerary tails, shorter and less distinct than the 
principal tail, are by no means uncommon ; but they generally 
appear quite suddenly, and as suddenly disappear in a few days, as 
if the stock of materials from which they were supplied had be- 
come exhausted. These secondary tails, by their periodical 
changes of position from the one side of the principal tail to the 
other, have made known the fact that the comets to which they 
belonged had a rotatory motion around the axis or central line of the 
tail. The same fact has been inferred from other phenomena, in 
the case of some other comets, as the great comet of 1811, and 
Halley's comet in 1835. 

554. The general position of the tail of a comet is nearly but 
not exactly in the prolongation of the line of the centres of the 
sun and head of the comet, or of the radius-vector of the comet. 
(See Fig. 97.) It deviates from this line on the side of the regions 
of space which the comet has just left ; and the angle of deviation, 
which, when the comet is first seen at a distance from the sun, is 
very small or not at all perceptible, increases as the comet ap- 
proaches the sun, and attains to its maximum value ■ soon after the 
perihelion passage ; after which it decreases, and finally, at a dis- 
tance from the sun, becomes insensible. For example, the angle 
of deviation of the tail of the great comet of 1811 attained to its 
maximum about ten days after the perihelion passage, and was 
ihen about 11°. In the case of the comet of 1664, the same angle 
about two weeks after the perihelion passage was 43°, and was 
then decreasing at the rate of 8° per day. 

The comet of 1823 might seem to present an exception to the 
general fact that the tail of a comet is nearly opposite to the sun ; 



PHYSICAL CONSTITUTION OF COMETS. 207 

but Arago has suggested that the probable cause of the singular 
phenomenon of a secondary tail, apparently directed towards the 
sun in the heavens, was that the earth was in such a position that 
the two tails, although in fact inclined to each other under a small 
angle, were directed towards different sides of the earth, and thus 
were referred to the heavens so as to appear nearly opposite. 

The same principle will serve to show that the deviation of the 
tail of a comet, from the position of exact opposition to the sun, 
may appear to be much greater than it actually is, by reason of the 
earth happening to be within the angle formed by the direction of 
the tail with the radius-vector prolonged. 

555. Comets are the most voluminous bodies in the solar sys- 
tem. The tail of the great comet of 1680 was found by Newton 
to have been, when longest, no less than 123,000,000 miles in 
length : according to Professor Peirce, the remarkable comet of 
1843, about three weeks after its perihelion passage, had a tail of 
over 200,000,000* miles in length. Other comets have had tails 
of from fifty to a hundred millions of miles in length. The heads 
of comets are usually many thousand miles in diameter. That of 
the comet of 181 1 had a diameter of 132,000 miles. Its envelope 
or nebulosity was 30,000 miles in thickness ; and the inner surface 
of this was no less than 36,000 miles distant from the centre of the 
nucleus. The head of the great comet of 1843 w T as about 30,§00 
miles in diameter. 

The nuclei of comets are in general only a few hundred miles in 
diameter : but according to Schroeter the nucleus of the comet of 
1811 had a diameter of 2600 miles ; and the nucleus of the comet of 
1843 seems to have been still greater. On the other hand, the 
comet of 1798 had a nucleus of less than 50 miles in diameter. 

It is important to observe that the dimensions of comets are sub- 
ject to continual variations. The tail increases as the comet ap- 
proaches the sun, and attains to its greatest size a certain time after 
the perihelion passage ; after which it decreases. The head, on 
the contrary, generally diminishes in size during the approach to 
the sun, and augments during the recess from him. The changes 
are often very sudden and rapid. 

PHYSICAL CONSTITUTION OF COMETS. 

556. The quantity of matter which enters into the constitution 
of a comet is exceedingly small. This is proved by the fact that 
the comets have had no influence upon the motions of the planets or 
satellites, although they have in many instances passed near these 
bodies. The comet of 1770, which was quite large and bright, 
passed through the midst of Jupiter's satellites, without deranging 
their motions in the least perceptible degree. Moreover, since this 
small quantity of matter is dispersed over a space of tens of thou- 

* According to later determinations 108,000,000 miles. 



208 OF COMETS. 

sands, or millions of miles (if we include the tail,) in linear extent, 
the nebulous matter of comets must be incalculably less dense than 
the solid matter of the planets. In fact, the cometic matter, with 
the exception perhaps of that of the nucleus, is inconceivably more 
rare and subtile than the lightest known gas, or the most evanescent 
film of vapor that ever makes its appearance in our sky ; for faint 
telescopic stars are distinctly visible through all parts of the comet, 
with, it may be, the exception of the nucleus in some instances, 
notwithstanding the great space occupied by the matter of the 
comet which the light of the star has to traverse. The matter of 
the tail of a comet is even more attenuated than that of the general 
mass of the nebulosity of the head ; but is apparently of the same 
nature, and derived from the head. The nucleus is supposed by 
some astronomers to be, in some instances, a solid, partially 
or wholly convertible into vapor, under the influence of the sun ; 
by others, to be in all cases the same species of matter as is in the 
nebulosity, only in a more condensed state ; and by others still, to 
be a solid of permanent dimensions, with a thick stratum of con- 
densed vapors resting upon its surface. Whichever of these views 
be adopted, it is a matter of observation that the nebulosity fre- 
quently receives fresh supplies of nebulous matter from the nu- 
cleus. It was the opinion of Sir William Herschel, and it has been 
the" more generally received notion since his time, that the nucleus 
of a comet is surrounded with a transparent atmosphere of vast 
extent, within which the nebulous envelope floats, as do clouds in 
the earth's atmosphere. But Olbers, and after him Bessel, con- 
ceives the nebulous matter of the head to be either in the act of 
flowing away into the tail under the influence of a repulsion from 
the nucleus and the sun, or in a state of equilibrium under the ac- 
tion of these forces and the attraction of the nucleus. 
• It is not yet definitively settled whether the cometic matter is 
self-luminous, or shines with the light received from the sun ; but 
it is the general opinion that it derives its light from the sun. 

CONSTITUTION AND MODE OF FORMATION OF THE TAILS OF COMETS. 

557. Upon this topic we may lay down the following postulates. 1. The gen- 
eral situation of the tail of a comet with respect to the sun, shows that the sun is 
concerned, either directly or indirectly, in its formation. The changes which take 
place in the dimensions of a comet, both in approaching the sun and receding from 
him, conduct to the same inference. 2. Since the tail lies in the direction of the 
radius-vector prolonged beyond the head, the particles of matter of which it is 
made up must have been driven off by some force exerted in a direction from the 
sun. 3. This force cannot emanate from the nucleus, for such a force would ex- 
pel the nebulous matter surrounding the nucleus in all directions, instead of one 
direction only. It is, however, conceivable that, as Olbers supposes, the nebulous 
matter is in the first instance expelled from the nucleus by its repulsive action, 
taking effect chiefly on the side towards the sun, and afterwards driven past the 
nucleus into the tail by a repulsion from the sun. 4. There seems, then, to be little 
room to doubt that the matter of the tail is driven off from the head by some force 
foreign to the comet, and taking effect from the sun outwards. 5. This force, 



FORMATION OF THE TAILS OF COMETS. 209 

whatever may be its nature, extends far beyond tbe earth's orbit. For comets 
have been seen provided with tails of great length, though their perihelion distance 
exceeded the radius of the earth's orbit, (e. g. the great comet of 1811.) Nothing 
can be predicated with certainty with respect to the law of variation of this force, 
but it is at least probable that, like all known central forces, it varies inversely as 
the square of the distance. 

558. Whatever may be the nature of the force in question ; whether it consists 
in an impulsive action of the sun's rays, as Euler imagined, or in a repulsion by 
the distant mass of the sun, consequent upon a polarity of the cometic particles 
induced by some action of the sun, as supposed by Olbers and Bessel, we will call 
it the repulsive force of the sun. Granting its existence, there are two modes in 
which we may conceive it to operate in forming the tail. We may suppose that 
it drives off the nebulous matter to greater and greater distances, *as its intensity 
increases, without destroying the original physical connection of the parts ; so that 
the tail and the head will always be revolving as one connected mass. Or we may 
conceive that it is continually detaching portions of the nebulosity, or turning them 
back if repelled by the nucleus, and repelling them to an indefinite distance into 
free space. The first mentioned conception is the theory which has generally pre- 
vailed hitherto ; but there seem to be good and sufficient reasons for rejecting it, 
and adopting the other in its stead. 1. There appears to be no satisfactory rea- 
son to be assigned why the force which expels the nebulous matter to the end of 
the apparent tail should not urge it still farther ; since the extremities of the tails 
of some comets are not so far removed from the sun as the heads of others, from 
which the nebulous matter is expelled by the same force. To account for the sup- 
posed limited extent of the actual tail, we are forced to suppose that the tendency 
of the particles to return to the nucleus increases as their distance from the nucleus 
and from one another increases ; which seems highly improbable. 2. Bessel has 
found that the nebulous matter of comets has no power to refract the light of a 
star, passing through it, whence he infers that there can be no molecular connec- 
tion between the particles. 3. It appears, by calculation, that in the case of the 
great comet of 1843, we cannot find either in the repulsive action of the sun upon 
the tail, or in the excess of attraction of the sun for the nearer parts of the comet, 
a force adequate to keep the tail continually opposite to the sun, and which at the 
same time will not sensibly alter the orbit, without making improbable suppositions 
as to the disproportion between the quantity of matter in the nucleus and tail.* 
4. In the case of such comets as that of 1843, and that of 1680, which come near 
the sun, the centrifugal force generated by the great velocity of rotation about the 
time of the perihelion passage would be so great as infallibly to dissipate the great- 
er part of the tail. At the time of the perihelion passage of either of these comets, 
the centrifugal force must have exceeded the gravity towards the nucleus at only 
a few hundred miles from the centre of gravity of the whole mass. 5. Whether we 
suppose the whole mass of the comet to be kept in rotation about its centre of grav- 
ity by the repulsion or by the attraction of the sun, the velocity of rotation, as it 
is constantly nearly equal to the angular velocity of revolution, must be on the in- 
crease up to the time of the perihelion passage. Now, this will not undergo any 
diminution after the perihelion passage, as the action of the force would tend to 
increase rather than diminish it, but the velocity of revolution will continually de- 
crease : it follows, therefore, that soon after the perihelion passage the velocity of 
rotation would exceed that of revolution, and continually more and more ; so that 
ere long the tail would inevitably be thrown forward of the line of the radius-vector 
prolonged, a situation in which the tail of a comet has never been seen. 

We here suppose the dimensions of the comet to remain the same. In point of 
fact, the apparent tail increases in length for a certain number of days after the 
perihelion passage. The tendency of this would be to diminish the velocitv of ro- 
tation ; but the supposed subsequent contraction of the tail to its original dimen- 
sions would restore the original velocity. 

In view of all that has now been stated, it seems highly probable that the tail 
and head of a comet do not form one connected body of matter, as has been gen- 
erally supposed ; but, on the contrary, that the tail is made up of particles of mat- 
ter continually in the act of flowing away at a very rapid rate from the head into 

* See Silliman's Journal, vol. xlvi, No. I, page 110, &c. 
27 



210 



OF COMETS. 



free space, under the action of the repulsive force of the sun, so called. According 
to this view, the tail which we see at any instant is the collection of all the parti- 
cles that have been emitted during a certain previous interval ; and at the end of 
every such interval we are looking at an entirely new tail. This theory of the con- 
stitution of the tails of comets is identical with Olbers' ; but, as we have seen (556), 
Olbers has also given a special theory of the constitution of the nebulosity of the 
heads of comets, of which nothing is here predicated. 

559. If a theory be true, it must furnish a satisfactory explanation of the facts 
and phenomena that fall within its scope. Let us examine the present theory from 
this point of view. In the first place, as respects the form of the tail, it is mani- 
fest from what has already been stated, (546,) that the sides of the tail must often 
diverge much more rapidly than the lines of action of the repulsive force of the 
sun upon the opposite parts of the head. Calculation shows this to have been the 
•case even in the comet of 1843. Now, according to Olbers' theory, this fact is a 
simple consequence of the supposed repulsive action of the nucleus. If we adopt 
Herschel's theory of the constitution of the head, we have apparently a sufficient 
cause for the same fact in the centrifugal force generated by the rotation of the tail, 
(553,) which we must suppose to be a consequence of a rotation of the head. 

560. In the next place, the increase in the length of the apparent tail as the 
comet approaches the sun, and until a certain time after the perihelion passage, 
may be naturally supposed to proceed from the emission of greater quantities of 
luminous matter in a given time, and a continued augmentation, up to the time of 
the perihelion passage, in the light received from the sun. The actual tail, it is to 
be observed, is really indefinite in length, and terminates, to us, where its matter 
becomes too much dispersed and too distant from the sun, the probable source of 
its light, to send us a perceptible light. 

561. Let us now see, in the third place, how the theory under consideration ac- 
counts for the situation and curvature of the tail. Let PCA (Fig. 99) be a portion 
of a comet's orbit, the sun being at S : and suppose a particle to be expelled in the 

Fig. 99. 




direction SAD, when the head is at A, and another particle to be driven off in the 
direction SBE, when the head is at B. Each particle will retain the orbitual mo- 
tion which obtained at the time of its departure, as it moves away from the sun ; 
and thus, when the comet has reached the point C, instead of being at any points 
D and E on the lines SAD and SBE, will be respectively at certain points a and b 
farther forward. The line Cba, which, when the comet is at C, is the locus of all 
the particles that have been emitted during the interval of time in which the comet 
has been moving over the arc AC, is the tail. We here suppose the head to be a 
mere point. If we conceive the particles to be continually emitted from the mar- 
ginal parts of the head, we shall have the hollow conical tail actually observed. It 
is easy to see that Cba, the line of the tail, must be a curved line concave towards 






NUMBER AND DISTRIBUTION OF THE FIXED STARS. 211 

ihe regions of space which the comet has left. Supposing the arc AC to be so 
small, or its curvature to be so slight that it may be considered as a straight line, 
and neglecting the change of the velocity in the orbit, C<z will be parallel to AD, 
and Cb parallel to BE, whence RCa = CSA, and RC6 = CSB. Thus the line 
joining any particle with the nucleus always makes an angle with the prolonga- 
tion of the radius-vector, equal to the motion in anomaly during the* interval 
that has elapsed since the particle left the head. It follows from this that, if 
we suppose the velocity of the particles to be continually the same, and the mo- 
tion in anomaly to be uniform, the deviations of the particles a and b from the 
line of the radius-vector SCR will be in the ratio of the distances Ca and Cb. But, 
in point of fact, the velocity increases with the distance, so that the curvature of 
the tail will be less than on the supposition just made. 

As to the amount of the deviation of the tail from the line of the radius-vector, 
it must depend upon the proportion between the velocity of the particles and the 
velocity of the head in its orbit : and it follows from the principle just established, 
that unless the velocities of emission augment as rapidly as the velocity of revolu- 
tion, the deviation in question will increase to the perihelion, and afterwards de- 
crease ; as it is in fact known to do. 

562. In support of Olbers' theory of a repulsion from the nucleus, it may be sta- 
ted, that the form of the nebulosity which this theory requires, was found by obser- 
vation to obtain in the case of the great comet of 1811, and also of Halley's Com- 
et in 1835.* 



CHAPTER XVIII. 



OF THE FIXED STARS 
THEIR NUMBER AND DISTRIBUTION OVER THE HEAVENS. 

563. The number of stars visible to the naked eye, in the entire 
sphere of the heavens, is from 6000 to 7000 ; of which nearly 
4000 are in the northern hemisphere ; but not more than 2000 can 
be seen with the naked eye at any one time at a given place. 
The telescope brings into view many millions, and every material 
augmentation of its space-penetrating power greatly increases the 
number. 

564. As to the number of stars belonging to each different mag- 
nitude, astronomers assign from 20 to 24 to the first magnitude, 
from 50 to 60 to the second, about 200 to the third, and so on ; 
the numbers increasing very rapidly as we descend in the scale 
of brightness ; the whole number of stars already registered down 
to the seventh magnitude, inclusive, amounting to 12,000 or 15,000.t 

The reason of this increase in the number of the stars, as we 
descend from one magnitude to another, is undoubtedly that in 
general the stars are less bright in proportion as their distance is 
greater; while the average distance between contiguous stars is 
about the same for one magnitude as for another. It is easy to 
see that upon these suppositions the number of stars posited at 

* See Silliman's Journal, vol. xlv. No. I, page 206. 
t HerschePs Outlines of Astronomy, p. 520. 



212 OF THE FIXED STARS, 

any given distance, and having therefore the same apparent mag- 
nitude, will be greater in proportion as this distance is greater, and 
thus as the apparent magnitude is lower. 

565. It is not to be understood that the classification of the star* 
into different magnitudes is made according to any fixed definite 
proportion subsisting between the degrees of apparent brightness 
of the stars belonging to different classes. Stars of almost every 
gradation of brightness, between the highest and the lowest, are 
met with. Those which offer marked differences of lustre, form 
the basis of the classification ; others, which do not differ very 
widely from these, are united to them. As a necessary conse- 
quence, there are some stars of intermediate lustre, which cannot 
be assigned with certainty to either magnitude. Thus, in the cata- 
logue of the Astronomical Society of London, 3 stars are marked 
as intermediate between the first and second magnitudes, and 29 
between the second and third. 

Different astronomers also not unfrequently assign the same star 
to different magnitudes. 

As to the proportions of light emitted from the average stars of 
the different magnitudes, according to the experimental comparisons 
of Sir Wm. Herschel, they are, from the first to the sixth magni- 
tude, approximately in the ratio of the numbers, 100, 25, 12, 6, 2, K 

566. With the exception of the three or four brightest classes.. 
the stars are not distributed indiscriminately over the sphere of the 
heavens, but are accumulated in far greater numbers on the borders 
of that belt of cloudy light in the heavens, which is called the 
milky way, and in the milky way itself, which the telescope shows 
to consist of an immense number of stars of small magnitude in 
close proximity. 

Herschel found that on a medium estimate a segment of the 
milky way, 15° long, and 2° broad, contained at least 50,000 stars 
of sufficient magnitude to be distinguished through his telescope.* 
According to this, taking its average breadth at 14°, the milky way 
must contain more than eight millions of stars. 

567. This great accumulation of stars in a zone of the heavens, 
encompassing the earth in the direction of a great circle, suggested 
to the mind of Herschel the idea that the stars of our firmament 
are not disseminated indifferently throughout the surrounding re- 
gions of space, but are for the most part arranged in a stratum, 
the thickness of which is very small in comparison with its breadth :. 

— the sun and solar sys- 
tem being near the mid- 
dle of the thickness. If 
S (Fig. 100) represents 
the place of the sun, it 
will be seen that upon 
this supposition the 

* A Newtonian reflecting telescope of 20 feet focus and nearly 19 in. in aperture. 




ANNUAL PARALLAX OF THE STARS. 



213 



number of stars in the direction SC of the thickness of the stratum 
-will be less than in any other direction, and that the greatest num- 
ber will lie in the direction of the breadth, as SB. On one side 
of the point S, the stratum is supposed to be divided for a cer- 
tain distance into two laminae, as Fig. 101. 
shown in the figure, which repre- 
sents a section of the supposed stra- 
tum. This supposition is necessary 
to account for the two branches, 
with a dark space between them, 
into which the milky way is divided 
for about one-third of its course. 

Hersehel undertook to gauge this stratum 
in various directions, on the principle that 
the distance through to its borders in any 
direction was greater in proportion as the 
number of stars seen in that direction was 
.greater. He thus found that its actual form 
was very irregular : its section, instead of 
being truly that of a segment of a sphere 
divided for a certain distance into two lami- 
nae, as represented in Fig. 100, having the 
form represented in Fig. 101. He estimated 
the thickness of the stratum to be less than 
160 times the interval between the stars, and 
the breadth to be nowhere greater than 1000 
times the same distance. He conceived that 
it extended in no direction a distance equal 
to the space-penetrating power of his tele- 
scope for individual stars, and. much less for 
collections of stars seen as nebulous spots. 

568. Sir John Hersehel conceives 
that the superior brilliancy and 
larger development of the milky 
way in the southern hemisphere, 
from the constellation Orion to that 
of Antinous, indicate that the sun 
and his system are at a distance from 
the centre of the stratum .in the di- 
rection of the Southern Cross, and 
that the central parts are so vacant 
of stars that the whole approximates 
to the form of an annulus. 




ANNUAL PARALLAX AND DISTANCE OF THE STARS. 



569. The Annual Parallax of a fixed star is the angle made by 
two lines conceived to be drawn, the one from the sun and the 
other from the earth, and meeting at the star, at the time the earth 
is in such part of its orbit that its radius-vector is perpendicular to 
the latter line ; or, in other words, it is the greatest angle that can 



214 OF THE FIXED STARS. 

be subtended at the star by the radius of the earth's orbit. Thus, 

let S (Fig. 102) be the sun, s a fixed star, and E the earth, in 

such a position that the radius-vector SE is perpendicular to Es 

Fig. 102. the line of direction of the star, 

^^f & then the angle SsE is the an- 

^^^/ nual parallax of the star s. 

^^^ yS 570. If the annual parallax 

E^^CT— _.y/-— -v>A of a star was known, we might 

jS ^^\ ./ ,,--''' easily find its distance from the 
/ /\ -''''' earth ; for in the right-angled tri- 

/ /.--' \ angle SEs we would know the 

s angle SsE and the side SE, and 

\ / we should only have to com- 

\ / pute the side Es. Now, if any 

^- — --^ of the fixed stars have a sensi- 

ble parallax, it could be detected by a comparison of the places of 
the star, as observed from two positions of the earth in its orbit, 
diametrically opposite to each other ; and accordingly, the atten- 
tion of astronomers furnished with the most perfect instruments,, 
has long been directed to such observations upon the places of 
some of the fixed stars, in order to determine their annual paral- 
lax. But, after exhausting every refinement of observation, they 
have not been able to establish that any of them have a measura- 
ble parallax. Now, such is the nicety to which the observations 
have been carried, that, did the angle in question amount to as- 
much as I", it could not possibly have escaped detection and uni- 
versal recognition. We may then conclude that the annual par- 
allax of the nearest fixed star is less than \" . 

571. Taking the parallax at 1", the distance of the star comes 
out 206,265 times the distance of the sun from the earth, or about 
20 billions of miles. The distance of the nearest fixed star must 
therefore be greater than this. A juster notion of the immense 
distance of the fixed stars, than can be conveyed by figures, may 
be gained from the consideration that light, which traverses the 
distance between the sun and earth in 8m. 18s., and would perform 
the circuit of our globe in J- of a second, employs more than three- 
years in coming from the nearest fixed star to the earth. 

According to Struve, the mo&t probable value of the paralbax of a star of the 
first magnitude is no more than about i" ; which would make its distance 5 times- 
greater than the above. determination. 

572. The statement made in Art. 570, that the annual parallax of the fixed stars 
has hitherto escaped certain detection, although truly representing the result of all 
the many efforts made to solve the great problem of the distance of the fixed stars,, 
until a very recent date, is at the present time (1845) no longer true. The paral- 
lax of one of the fixed stars is now believed to have been determined by BesseL 
This is the star 61 Cygni.* It is a star of about the 6th magnitude, barely visible 
to the naked eye. When viewed through a telescope it is seen to consist of two 
stars of nearly equal brightness, at a distance from each other of about 16". These 

* R. A. 314° 52', Dec.N. 37° 56'. 



DISTANCE OF THE STARS. 



215 



stars have a motion of revolution around each other, and the two move together 
at the same rate, of 5".3 per year, as one star, along the sphere of the heavens. It 
is hence inferred that they are bound together into one system by the principle of 
gravitation, and are at pretty nearly the same distance from the earth. The great 
proper motion of this double star, as compared with other stars, led to the suspicion 
that it was nearer than any other ; and thus to attempts to determine its parallax. 
The principle of Besstl's method is to find the difference between the parallaxes of 
the star 61 Cygni, and some other star of much smaller magnitude, and therefore 
supposed to be at a much greater distance, seen in as nearly the same direction as 
possible. This difference will differ from the absolute parallax of the double star 
by only a small fraction of its whole amount. It was found by measuring with a 
position micrometer (76) the annual changes in the distance of the two stars, and 
in the position of the line joining them. To make it evident that such changes 
will be an inevitable consequence of any difference of parallax in the two stars, 
conceive two cones having the earth's orbit for a common base, and their vertices 
respectively at the two stars, and imagine their surfaces to be produced past the 
stars until they intersect the heavens. The intersections will be ellipses, but, by 
reason of the different distances of the two 
stars, of different sizes, as represented in Fig. 
103 ; and they will be apparently described by 
the stars in the course of one revolution of the 
earth in its orbit. The two stars will always 
be similarly situated in their parallactic ellip- 
ses : thus, if one is at A the other will be at 
a ; and after the earth has made ore-quarter 
of a revolution, they will be at B and b ; and af- 
ter another quarter of a revolution at C and c, 
&c. Now it will be manifest, on inspecting 
the figure, the ellipses being of unequal size, 
that the line of the stars will be of unequal 
lengths, and have different directions in the 
different situations of the stars. 

A much smaller angle of parallax may be 
found, with the same degree of certainty, by 
this indirect method than by the direct process 
explained in Art. 570 ; for, since the two stars are seen in pretty nearly the same 
direction, they will be equally affected by refraction and aberration ; and since it 
is only the relative situations of the two stars that are measured, no allowance has 
to be made for precession and nutation, or for errors in the construction or adjust- 
ment of the instrument. It is therefore independent of the errors that are inevita- 
bly committed in the determination of these several corrections, when it is at- 
tempted to find directly the absolute parallax, by observing the right ascension and 
declination at opposite seasons of the year. The measurements made with the 
micrometer in the hands of the most accurate observers, may be relied on as exact 
to within a small fraction of 1". 

For the sake of greater certainty Bessel made the measurements of parallactic 
changes of relative situation between the star 61 Cygni, and two small stars in- 
stead of one, — the middle point between the two members of the double star being 
taken for the situation of this star. He found the difference of parallax to be for 
the one star 0".3584, and for the other star 0".3289 : and assuming the absolute 
parallax of the two stars to be equal, found for the most probable value of the dif- 
ference of parallax 0".3483. Whence he calculated the distance of the star 61 
Cygni to be 592,200 times the mean distance of the earth from the sun ; a distance 
which would be traversed by light in 9$ years. (See Note XIV.) 

573. The amount of light received from the same body at differ- 
ent distances varies inversely as the square of the distance. Hence, 
if we admit the light of a star of each magnitude to be half that of 
one of the next higher magnitude, a star of the first magnitude 
would have to be removed to 360 times its distance, to appear no 
brighter than one of the eighteenth. Accordingly, if the difference 




216 OF THE FIXED STAH». 

in the apparent magnitude of the stars arises for the most part 
from a difference of distance, (which is the more probable suppo- 
sition,) there must be a multitude of stars visible in telescopes, 
the light of which has taken at least one thousand years to reach 
the earth. 

A calculation based upon the power of large telescopes to aug- 
ment the amount of light received from the stars, in connection with 
the well-known law of diminution of the light received as the dis- 
tance increases, conducts to about the same result. 

NATURE AND MAGNITUDE OF THE STARS. 

574. The vast distance at which the fixed stars are visible, and 
shine with a light not much inferior to the planets, leaves no room 
to doubl, that they are all suns like our own. If it should be con- 
jectured that some of the fainter stars might be bodies shining 
by reflected light, like the planets, the answer is, that if we were 
to suppose the existence of opake bodies, at the distance of the 
stars, so inconceivably vast in their dimensions as to send a sensi- 
ble light to the eye, if illuminated to the same degree as the plan- 
ets, the stars of the smaller magnitudes are, with the exception 
perhaps of the members of some of the double stars, too remote 
from the brighter ones to receive sufficient light from them ; for, 
the smallest measurable space in the field of the largest telescopes 
is, at the distance of the nearest star, as large as or larger than the 
earth's orbit. It is perhaps possible, that some of the faintest 
members of some of the double stars, as surmised by Sir John 
Herschel, may shine by reflected light. 

575. To be able to determine the magnitude of a star, we must 
know its distance, and also its apparent diameter. Now the dis- 
tance of only one star has, as yet, been found ; and the discs of 
all the stars, even in the most powerful telescopes, are altogether 
spurious ; so that in no instance have we the data, nor have we 
reason to expect that they will be hereafter obtained, for determin- 
ing with certainty the magnitude of a fixed star. 

We may infer, however, from the intensity of their light, that it 
is highly probable that some at least of the stars are as large as, 
or even larger than the sun. It has been calculated from the re- 
sults of photometrical experiments made by Dr. Wollaston, on the 
relative quantity of light received from Sirius and the sun, that if 
the sun were removed to the distance of 20 billions of miles, which 
is known to be less than the distance of any of the stars, he would 
not send to us so much as half the quantity of li^ht actually re- 
ceived from Sirius. 

576. Although there are not sufficient data for calculating the magnitude of the 
star 61 Cygni, there are for ascertaining its mass. This element results from the 
distance and the motion of revolution of the two members of the double star about 
each other. Bessel finds it to be less than half of the sun's mass. According to 



VARIABLE STARS. 217 

this result the sun, as seen from this star, should appear as a star of about the fifth 
magnitude. 

VARIABLE STARS. 

577. A number of the fixed stars are subject to periodical 
changes of brightness, and are hence called Variable Stars, or 
Periodical Stars. One of the most remarkable of the variable 
stars is the star Omicroji, in the constellation Cetus. From being 
as bright as a star of the second magnitude, it gradually decreases 
until it entirely disappears ; and, after remaining for a time invisi- 
ble, reappears, and gradually increasing in lustre, finally recovers 
its original appearance. The period of these changes is 332 days. 
It remains at its greatest brightness about two weeks, employs 
about three months in waning to its disappearance, continues invi- 
sible for about five months, and during the remaining three months 
of its period increases to its original lustre. Such is the general 
course of its phases. It does not, however, always recover the 
same degree of brightness, nor increase and diminish by the same 
gradations. It is related by Hevelius, that in one instance it re- 
mained invisible for a period of four years, viz. from October, 1672, 
to December, 1676.* A similar phenomenon has been noticed in 
the case of another variable star, viz. the star x Cygni. It is sta- 
ted by Cassini to have been scarcely visible throughout the years 
1699, 1700, and 1701, at those times when it ought to have been 
most conspicuous. On the other hand, a variable star, situated in 
the Northern Crown, sometimes continues visible for several years 
without any apparent change, and then resumes its regular varia- 
tions. 

578. The greater number of variable stars undergo a regular 
increase and diminution of lustre, without ever, like the star just 
noticed, becoming entirely invisible. The star Algol, or /3 Perseii, 
is a remarkable variable star of this description. For a period of 
2d. 14h. it appears as a star of the second magnitude, after which 
it suddenly begins to diminish in splendor, and in about 3^ hours 
is reduced to a star of the fourth magnitude. It then begins again 
to increase, and in 3| hours more is restored to its usual bright- 
ness, going through all its changes in 2d. 20h. 48m. t 

579. There are also a number of double stars, one or both of 
the members of which are variable ; as y Virginis, s Arietis, 
£ Bootis, &c. 

580. Two general facts have been noticed with respect to the 
variable stars, which are worthy of remark, viz. that the color of 
their light is red, and that their phase of least light lasts much long- 
er than that of their greatest light. The star Algol, which is white, 
is said to be the only variable star whose light is not of a reddish 

* Herschel's Treatise on Astronomy, p. 356. t Ibid. 357. 
28 



218 OF THE FIXED STARS. 

color. The same star also presents an exception to the other gen- 
eral fact just noticed. (See Note XV.) 

581. There are also some instances on record of temporary 
stars having made their appearance in the heavens ; breaking forth 
suddenly in great splendor, and without changing their positions 
among the other stars, after a time entirely disappearing. One of 
the most noted of these is the star which suddenly shone forth with 
great brilliancy on the 11th of November, 1572, between the con- 
stellations Cepheus and Cassiopeia, and was attentively observed 
by Tycho Brahe. It was then as bright as any of the permanent 
stars, and continued to increase in splendor till it surpassed Jupiter 
when brightest, and was visible at mid-day. It began to diminish 
in December of the same year, and in March, 1574, it entirely dis- 
appeared, after having remained visible for sixteen months, and 
has not since been seen.* 

It was noticed that while visible the color of its light changed from white to yel- 
low, and then to a very distinct red ; after which it became pale, like Saturn. 

In the years 945 and 1264, brilliant stars appeared in the same 
region of the heavens. It is conjectured from the tolerably near 
agreement of the intervals of the appearance of these stars and 
that of 1572, that the three may be one and the same star, with a 
period of about 300 years. The places of the stars of 945 and 
1264 are, however, too imperfectly known to establish this with 
any degree of certainty. 

Besides these three temporary stars, several others have made their appearance, 
viz. one in the year 125 B. C, seen by Hipparchus; another in 389 A. D., in the 
constellation Aquila ; a third in the 9th century, in Scorpio ; a fourth in 1604, in 
Serpentarius, seen by Kepler; and a fifth in 1670, in the Swan. 

582. What is no less remarkable than the changes we have no- 
ticed, several stars, which are mentioned by the ancient astrono- 
mers, have now ceased to be visible, and some are now visible 
to the naked eye which are not in the ancient catalogues. 

583. The most probable explanation of the phenomenon of variable stars, is, that 
they are self-luminous bodit s rotating upon axes, like the sun, and having like him 
spots upon their surface, but vastly larger and more permanent. By the rotation 
these spots are brought periodically around on the side towards the earth, and ac- 
cording to their size occasion a diminution of the light of the star, or make it en- 
tirely to disappear. In the case of the star Algol, however, as suggested by Good- 
ricke, the phenomena are precisely such as would result from the periodical inter- 
position of an opake body revolving around it. In those cases in which the period 
of the diminution of the light is a large fraction of the entire period of the star, 
(580,) as well as those in which there are occasional interruptions in the regular 
recurrence of the phenomena, (577,) the supposition of the interposition of an opake 
body will not answer. (See ~Note XVI.) 

584. Temporary stars are most probably suns which have entirely intermitted 
the evolution of light for a long period of time, and then burst forth anew with a 
sudden and peculiar splendor. Laplace conjectured that they might be the confla- 
grations of distant worlds ; but it seems very questionable whether the conflagra- 

* Herschel's Treatise on Astronomy, p. 359. 



DOUBLE STARS. 219 

tion of even, an entire system of planets would furnish as much light as the sun at 
its centre ; and the large and permanent spots on the surface of the variable stars 
would seem to render it probable that some suns have become, for a time, entirely 
extinct. In support of this theory, that temporary stars are the temporary revival 
of extinct suns, we have the fact said to have been recently discovered by Bessel, 
that there are opake bodies in space of the size of suns. It is stated that this 
distinguished astronomer has ascertained, from a discussion of the most accurate 
observations that have been made upon these stars, that the proper motions of the 
two stars Sirius and Procyon deviate sensibly from uniformity ; whence he infers 
that they must each be revolving about some large non-luminous body in their vi- 
cinity, and are thus double stars, one of the members of which is non-luminous, 

DOUBLE STARS. 

585. Many of the stars which to the naked eye appear single, 
when examined with telescopes are found to consist of two (in 
some instances three or more) stars in close proximity to each oth- 
er. These are called Double Stars, or Multiple Stars. (See Fig. 
104.) This class of bodies was first attentively observed by Sir 
William Herschel, who, in the years 1782 and 1785, published 

Fig. 104. 




Castor. y Leonis. Rigel. Pole-star. 11 31onoc. £ Cancri. 

catalogues of a large number of them which he had observed. The 
list has since been greatly increased by Professor Struve, of Dor- 
pat, Sir J. F. W. Herschel, and other ob servers, and now amounts 
to several thousand. 

5S6. Double stars are of various degrees of proximity. In a great 
number of instances, the angular distance of the individual stars is 
less than 1", and the two can only be separated by the most pow- 
erful telescopes. In other instances, the distance is \' and more, 
and the separation can be effected with telescopes of very moder- 
ate power. They are divided into different classes or orders, ac- 
cording to their distances ; those in which the proximity is the 
closest forming the first class. 

587. The tw T o members of a double star are generally of quite 
unequal size. (See Fig. 104.) But in some instances, as that of 
'the star Castor, they are of nearly the same magnitude. Double 

stars occur of every variety of magnitude. 

588. It is a curious fact, that the two constituents of a double star in numer- 
ous instances shine with different colors; and it is still more curious that these 
colors are in general complementary to each other. Thus, the larger star is 
usually of a ruddy or orange hue, while the smaller one appears blue or green. 
This phenomenon has been supposed to be in some cases the effect of contrast ; 
the larger star inducing the accidental color in the feebler light of the other. Sir 
John Herschel cites as probable examples of this effect the two stars i Cancri, and 
y Andromeda?. But it is maintained by Nichol that this explanation cannot be 
admitted ; for, if true, it ought to be universal, whereas there are many systems 
similar in relative magnitudes to the contrasted ones, in which both stars are yel- 
low, or otherwise belong to the red end of the spectrum. Again, if the blue or 



220 OF THE FIXED STARS. 

violet color were the effect of contrast, it ought to disappear when the yellow star 
is hid from the eye ; which, however, it does not do. Thus, the star (3 Cygni con- 
sists of two stars, of which one is yellow, and the other shines with an intensely 
blue light ; and when one of them is concealed from view by an interposed slip of 
darkened copper, the other preserves its color unchanged. The color, then, of 
neither of the stars can be accidental. 

It may be remarked in this connection, that the isolated stars also shine with 
various colors. For example, among stars of the first magnitude, Sirius, Vega, 
Altair, Spica are white, Aldebaran, Arcturus, Betelgeux red, Capella and Procy- 
on yellow. In smaller stars the same difference is seen, and with equal distinct- 
ness when they are viewed through telescopes. According to Herschel, insulated 
stars of a red color, almost as deep as that of blood, occur in many parts of the 
heavens, but no decidedly green or blue star has ever been noticed unassociated 
with a companion brighter than itself. 

589. Sir William Herschel instituted a series of observations 
upon several of the double stars, with the view of ascertaining 
whether the apparent relative situation of the individual stars expe- 
rienced any change in consequence of the annual variation of the 
parallax of the star. With a micrometer adapted to the purpose, 
(76,) he measured from time to time the apparent distance of the 
two stars, and the angle formed by their line of junction with the 
meridian at the time of the meridian passage, called the Angle of 
Position. Instead, however, of finding that annual variation of 
these angles, which the parallax of the earth's annual motion would 
produce, he observed that, in many instances, they were subject to 
regular progressive changes, which seemed to indicate a real mo- 
tion of the stars with respect to each other. After continuing his ob- 
servations for a period of twenty-five years, he satisfactorily ascer- 
tained that the changes in question were in reality produced by a 
motion of revolution of one star around the other, or of both around 
their common centre of gravity ; and in two papers, published in 
the Philosophical Transactions for the years 1803 and 1804, lie 
announced the important discovery that there exist sidereal sys- 
tems composed of two stars revolving about each other in regular 
orbits. These stars have received the appellation of Binary Stars, 
to distinguish them from other double stars which are not thus 
physically connected, and whose apparent proximity may be occa- 
sioned by the circumstance of their being situated on nearly the 
same line of direction from the earth, though at very different dis- 
tances from it. Similar stars, consisting of more than two consti- 
tuents, are called Ternary, Quaternary, &c. 

590. Since the time of Sir W. Herschel, the observations upon 
the binary stars have been continued by Sir John Herschel, Sir 
James South, Struve, Bessel, Madler, and other astronomers : ac- 
cording to Madler, the number of known binary and ternary stars 
is now about 250. Every year materially increases the list ; and 
will probably continue to do so for some time to come : for, while 
the changes of relative situation are in some instances exceedingly 
slow, the actual number of such systems is probably a large frac- 
tion of the whole number of double stars ; at least, if we confine 



DOUBLE STARS. 221 

our attention to double stars whose constituents are within J/ of 
each other. This may be inferred from the fact, that the num- 
ber of such double and multiple stars actually observed, which 
amounts to over 3000, is at least ten times greater than the num- 
ber of instances of fortuitous juxtaposition that would obtain on the 
supposition of a uniform distribution of the stars. Besides, there 
is a number of double stars not yet discovered to have a motion of 
revolution, which still give indications of a physical connection, 
Thus, their constituents are found to have constantly the same 
proper motion in the same direction ; showing that they are in all 
probability moving as one system through space. 

From the observations made upon some of the binary. stars, as- 
tronomers have been enabled to deduce the form of their orbits, 
and approximately the lengths of their periods. The orbits are 
ellipses of considerable eccentricity. The periods are of various 
lengths, as will be seen from the following enumeration of those 
which are considered as the best ascertained : a Coronse 608 years ; 
61 Cygni 540 years ; a Geminorum 232 years ; y Yirginis 182 
years ; 3062 Struve 95 years ; p Ophiuchi 93 years ; X Ophiuchi 
88 years ; w Leonis 83 years ; \ Ursas Majoris 60 years ; £ Can- 
cri 59 years ; v\ Coronas 43 years ; £ Herculis 31 years. Fig. 105 
represents a portion of the apparent orbit of the double star y Vir- 
ginis, and shows the relative positions of the two members of the 

Fig. 105. 







double star in various years. At the time of their nearest approach? 
in 1836, the interval between them was a fraction of 1", and they 
could not be separated by the best telescopes, with a magnifying 
power of 1000. Since then their distance has been continually 
increasing. In 1844 it amounted to 2", and a power of from 200 
to 300 was sufficient to separate them. The orbit represented in 
the figure is the stereographic projection of the true orbit on a 
plane perpendicular to the line of sight. (See Note XVII.) 



222 OF THE FIXED STARS. 

The actual distance between the members of a binary star has 
been found only for the star 61 Cygni. Bessel makes it for this 
star about two and a half times the distance of Uranus from the sun. 

591. It is important to observe, that the revolution of one star 
around another is a different phenomenon from the revolution of a 
planet around the sun. It is the revolution of one sun around an- 
other sun ; of one solar system around another solar system ; or 
rather of both around their common centre of gravity. We learn 
from it the important fact, that the fixed stars are endued with the 
same property of attraction that belongs to the sun and planets. 

PROPER MOTIONS OF THE STARS. 

592. It has already been stated (181) that the fixed stars, so 
called, are not all of them rigorously stationary. By a careful 
comparison of their places, found at different times with the accu- 
rate instruments and refined processes of modern observation, it 
has been found that great numbers of them have a progressive mo- 
tion along the sphere of the heavens, from year to year. The ve- 
locity and direction of this motion are uniformly the same for the 
same star, but different for different stars. The star which has 
the greatest proper motion of any observed, is the double star 61 
Cygni. During the last fifty years it has shifted its position in the 
heavens 4' 23" ; the annual proper motion of each of the individ- 
ual stars being 5". 3. Among isolated stars, jx Cassiopeia? has the 
greatest proper motion. It changes its place 3". 74 every year. 
The proper motions of some of the stars are either partially or en- 
tirely attributable to a motion of the sun and the whole solar sys- 
tem in space ; but the motions of others cannot be reconciled with 
this hypothesis, and must be regarded as in all probability indica- 
tive oi a real motion of these bodies in space. (See Note XVIII.) 

593. The first successful attempt to explain the proper motions 
of the fixed stars on the hypothesis of a motion of the solar system 
through space, was made by Sir William Herschel. After a care- 
ful examination of these motions, he conceived that the majority 
of them could be explained on the supposition of a general recess 
of the stars from a point near that occupied by the star X Herculis 
towards a point diametrically opposite. Whence he inferred that 
the sun with its attendant system of planets was moving rapidly 
through space in a direction towards this constellation. Doubt has 
since been thrown upon these conclusions by Bessel and other as- 
tronomers ; but they have quite recently been decisively re-estab- 
lished by M. Argelander, of Abo. The investigations of Arge- 
lander, which were communicated to the Academy of St. Peters- 
burgh in 1837, have since been confirmed by M. Otho Struve, of 
the celebrated Pulkova Observatory. 

Combining the determinations of these two astronomers, we find 
the most probable situation of the point towards which the sun's 






CLUSTERS OF STARS. NEBULA. 223 

motion is directed to be as follows : R. A. 259° 9', Dec. N. 34° 
36'. The point in question is situated in the constellation Hercules, 
near the star a, (No. 68 in the Catalogue of the Astronomical So- 
ciety,) and about 10° from the point first supposed by Herschel. 

594. O. Struve finds that for a star situated at right angles to the direction 
of the sun's motion, and placed at the mean distance of the stars of the .first 
magnitude, the annual angular displacement due to the sun's motion is 0".339, 
(with a probable error of 0".025.) So that, if we assume, according to the best 
determinations, 0".211 for the hypothetical value of the parallax of a star of the 
first magnitude, it follows that at the distance of the star supposed the annual 
motion of the sun subtends an angle about once and a half (1.606) greater than 
the radius of the earth's orbit: which makes it about 150,000,000 of miles. This 
is at the rate of about 4j miles per second. 

595. The above angle of 0".339 is about the greatest annual displacement which 
a star can experience in consequence of the sun's motion. Whence it appears that 
the whole of the proper motion of any star which is over and above this amount 
must certainly be due to a real motion in space. Thus, in the case of the star 61 
Cygni, at least 5" of its annual proper motion (5".23) results from an actual mo- 
tion in space. This is 14.3 times greater than the parallax of this star, (0".35.) 
Accordingly if we suppose the direction of its motion to be perpendicular to its line 
of direction from the sun or earth, its annual motion is 14.3 times greater than the 
radius of the earth's orbit, or at the rate of 43 miles per second. As we have 
no means of ascertaining the actual direction of its motion, it is impossible to 
discover how much it exceeds this determination. 

596. By comparing the particular motions presented by stars of different class- 
es with the motion of the solar system, viewed perpendicularly at the distance of 
a star of the first magnitude, as above given, it is found that the former, at the 
mean, are 2.4 times greater than that of the sun ; whence it follows that this lu- 
minary may be ranked among those stars which have a comparatively slow mo- 
tion in space 

CLUSTERS OF STARS.— NEBULAE. 

597. A great number of spaces are discovered in the heavens 
which are faintly luminous, and snine with a pale white light. 
These are called Nebula. Some are visible to the naked eye, but 
the greater number cannot be seen without the aid of a good tele- 
scope. On applying to them telescopes of great power, they are 
found for the most part to consist of a multitude of small stars, dis- 
tinctly separate, but very near each other, and more or less con- 
densed towards the centre. 

598. There are also clusters of stars in close proximity, dis- 
persed here and there over the sphere of the heavens, which are 
seen to be such with the naked eye, or with telescopes of only 
moderate power. One of the most conspicuous of these clusters 
is that called the Pleiades. 

To the unaided sight it appears to consist of six or seven stars, 
but a telescope even of moderate power exhibits within the space 
they occupy fifty or sixty conspicuous stars. The constellation 
called Coma Berenices, is another group, more diffused, and com- 
posed of larger stars. 

In the constellation Cancer there is a luminous spot, or nebula, 
called P?'cEsepe, or the bee-hive, which a telescope of moderate power 
resolves entirely into stars. In Perseus is another spot crowded 
with stars, which become separately visible with a good telescope. 



224 OF THE FIXED STARS. 

599. A large number of nebula are met with, indifferent parts 
of the heavens, which offer no appearance of stars, even when ex- 
amined with telescopes of the highest power. A very great diver- 
sity of form and aspect obtains among them. One of the most 
prominent is that near the star v in Andromeda. It is visible to the 
naked eye, and has often been mistaken for a comet. (See Fig. 106. N 

Fig. 106 




600. The number of nebulas at present known is about 3000. 
Although they occur in almost every part of the heavens, they are 
the most abundant in a zone perpendicular to the milky way, and 
of about the same breadth, and whose general direction is not very 
remote from that of the equinoctial colure ; and are particularly 
numerous where it crosses the constellations Virgo, Coma Bereni- 
ces, and the Great Bear. They are, for the most part, beyond the 
reach of any but the most powerful instruments. 

They are divided by Sir William Herschel into six different 
classes, as follows : 

(1.) Resolved Nehulcz ; that is, nebulae seen in the telescope to be clusters of 
stars. Of these some are globular in their form, and others of an irregular figure 
Yi s , 107. (See Fig. 107.) The latter ara 

less rich in stars, and less con. 
densed towards the centre than 
the globular clusters. They are 
also less definite in their outline. 
It is possible that they may be in 
the act of condensing, as Herschel 
supposed, and destined in the pro- 
cess of ages to form truly globu- 
lar clusters. This idea seems to 
be supported by the fact of the 
occurrence of a regular gradation 
of clusters, from one which seems 
to be only a space, of an irregular 
and ill-defined outline, somewhat 
more rich in stars than the sur- 
rounding regions, to the perfectly 
defined and isolated globular clus- 
ter highly condensed at the cen 




NEBULA. 



225 



tre. Globular clusters appear in telescopes of only moderate power, as small, round, 
or oval nebulous specks, resembling a comet without a tail. The number of stars 
which they contain is to be told only by thousands and tens of thousands ; although 
.their apparent size does not exceed the ~th part of the moon's disc. 

(2.) Resolvable Nebula ; or such as give indications that they are clusters of 
stars, and that they are in their nature resolvable into stars, although the power 
of the telescope is not yet sufficient to accomplish this. In telescopes of the high- 
est power they present the same appearance as the resolved globular clusters in 
telescopes which do not show their individual stars. Many of them have the as- 

Fiff. 108. 





pect of these globular clusters just before they are resolved, and which has been 
characterized by the phrase star-dust. They are of a round or oval form ; and are 
doubtless real clusters too distant to show either their irregular edges or their indi- 
vidual stars. (See Fig. 108.) 

(3.) Nebula Proper ; or which offer no appearance of stars, and are supposed 
to be actual masses of nebulous matter. Their nebulous constitution is inferred, 

Fig. 109. 




Nebula in Orion. 
1st, from their unique appearance, which is often quite different from that of the 
resolvable nebulae, and unlike what might be supposed to arise from an accumula- 
tion of stars. (See Fig. 109.) 2d. From their manifest physical connection with 

29 



226 OF THE FIXED STARS 

individual stars much superior to them in. brightness. The evidence of this physi- 
cal connection is found, in some instances, in an apparent condensation upon the 
star ; in others in the fact that the substance of the nebula, whatever it may be, 

Fig. 110. 




has apparently vacated the surrounding region? of space and accumulated about 
certain stars : (see Fig. 110 :) and in others still, in the circumstance of its being 
apparently drawn towards certain stars. (See Fig. 110.) 3. Another argument to 
the same point is, that though many of them are seen in telescopes of moderate 
power, and some with the naked eye, they are not only not resolved into stars by 
the largest telescopes, as other nebulae of the same brightness are, but do not like 
these assume a different appearance, farther than that they grow brighter, as the 
illuminating power of the telescope increases. 

Fig. 111. 



They present the greatest variety of forms, and occur in every stage of apparent 
condensation, from rude amorphous masses of almost equally diffused nebulous 
matter, to masses in which the condensation has progressed so far that a star is, 
to all appearance, beginning to be formed at the centre. The latter class have 
received a distinctive name, and will soon be particularly noticed. The condensa- 



DISTANCE OF NEBULA. 



227 



ition is often going on in the same mass upon several lines or points ; and masses 
are formed, presenting, in a regular gradation, all the varieties of appearance, 
which a mass, breaking up into parts by condensation upon points or lines, would 
assume down to the time of complete separation. The nebula? in which the con- 
densation appears to be upon one point or line, are round or oval in their figure. 
But some are long and spindle-shaped, while others are perfectly circular, Avery 
few of the round nebulas are annular. (Fig. 111.) A conspicuous example of 
this singular class of nebulas may be seen with a telescope of moderate power 
midway between the stars j8 and y Lyras. By far the greater portion of the nebulas 
proper are round. 

(4.) Planetary Nebula ; or nebulas which have an appearance similar to the plan- 
ets ; being round, of an equable light throughout, and often perfectly definite in their 
outline. (See Fig. 112.) The uniformity of their light seems Yig. 112. 

to indicate that it proceeds altogether from the surface of some 
spherical body ; and therefore that the body, if it be a collec- 
tion of nebulous matter, is of the form of a spherical shell : or 
else that it is derived from a bed of stars of uniform thick- 
ness. The latter supposition seems to be the more probable one, 
and is moreover now known to be true in some instances ; for 
Lord Rosse has succeeded in resolving one of the planetary ne- 
bulas of Sir J. Herschel's catalogue, (viz. Fig. 49 ;) and has discovered another 
(Fig. 45 of the catalogue) to be an annular nebula. 

The largest planetary nebula occurs in the Swan, and is nearly 15' in diameter. 

(5.) Stellar Nebula; that is, nebulas so much condensed at the centre as to offer 
fthe appearance of a star there seen through the surrounding nebulous mass. (See 

Fig. 112.* 





Fig. 112. a ) In some instances the condensation is gradual, in others sudden. A 
good example of stellar nebulae occurs to the south of the star /? Ursas Majoris. 

(6.) Nebulous Stars; or stars distinctly seen to be such, surrounded by their 
nebulous atmospheres. On the supposition of progressive condensation, they 
would seem to be stellar nebulas in a more advanced state. (See Fig. Il2. a ) 

Among stellar nebulas and nebulous stars there exist particular nebulas in every 
■stage of apparent condensation, from the slightest appearance of a star at the cen- 
tre to a perfect star surrounded with the faintest nebulous haze. (See Note XIX.) 

DISTANCE AND MAGNITUDE OF NEBULiE. 

601- Herschel undertook to estimate the distance of resolved 
nebulae, by noting the space-penetrating power of the telescope 
which first succeeded in revealing their distinct stars. According 
to his determinations, therefore, the most remote of the resolved 
nebulae are at the same distance as the most remote of the isolated 
stars discerned in his large telescope ; and thus at least 1000 times 
the distance of the nearest and brightest stars. The others are 
distributed at the same variety of distance as tile telescopic stars. 

602. As to the actual dimensions of these clusters, if we sup- 
pose the distance of none of them to be more than 1000 times the 
distance of a star of the first magnitude, a globular cluster whose 



22S OF THE FIXED STARS. 

apparent diameter is 10' cannot have a real diameter of more tnaffi 
three stellar intervals. . At a distance some 5 times greater, such 
a cluster would contain several thousand stars as remote from each 
other as is the nearest fixed star from our sun. 

603. It is to be supposed that the resolvable nebulae are in gene- 
ral posited beyond the region of resolved nebulae and visible stars. 
The nearest of them are on the very confines of this region. We 
may form some estimate of the probable distance of the most re- 
mote of these objects, by calculating how much farther a cluster, 
ascertained as above, (601,) to be at 1000 times the distance of 
the nearest star, and which is just discerned as a whitish speck by 
a telescope of the space-penetrating power 20, would have to be 
removed to have the same appearance in a telescope whose power 
is 200, (which is less than the power of the largest telescope.) It 
is plain that it would have to be removed 10 times farther, or to 
10,000 times the distance of the nearest isolated stars. 

This calculation supposes, however, that the number of stars in the most remote 
resolvable nebulae is no greater than in the most distant resolved nebulae. If we 
suppose the number to be greater in any ratio, the distance will be increased in the 
proportion of the square root of the same ratio. Thus suppose the number of stars 
in the remotest resolved nebulae to be 10,000, and that the most distant of the re- 
solvable nebulae contains a number 1000 times greater, or 10,000,000, (which is 
not far from the estimate of the number of stars in the stratum of the milky way,) 
(566). The distance calculated above will be increased about 30 times, that is ? 
will be no Jess than 300,000 times a stellar interval — a distance so enormous that 
light would employ 1,000,000 of years in traversing it. Some astronomers make 
the probable distance of stars of the lowest magnitude about three times less than 
we have taken it, which would make the distance just calculated less in the same 
proportion. Herschel, on the other hand, makes it about twice as great. If the 
bed of stars to which our sun belongs were viewed at this distance, it would sub- 
tend an angle of about 10', and appear about y^b- of the size of the moon's disc. 
It seems probable, a priori, that other similar beds of stars, to that in which our 
sun is posited, occur in the profundities of space. If this is the case they must 
then be visible at the enormous distance just stated, unless there be a limit to the 
known law of the propagation of light. 

604. It appears then that clusters of stars are distributed through- 
out space at every variety of distance, from that of stars of about 
the 4th magnitude to an unknown limit beyond the reach of the 
most powerful telescopes : and that the telescope succeeds in dis- 
tinctly resolving only those which are posited within the region of 
the isolated stars discernible through it. The more distant ones 
appear in it as spots of nebulous light, and occupy the fields of 
space which extend from say 1000 times the distance of stars of 
the first magnitude to at least 10,000 times this distance. 

605. As respects the nebulae proper, we may form some esti- 
mate of the distance of some of them by noting the magnitude of 
the stars with which they seem to be connected. In this way, for 
example, it is found that the remarkable nebula in Orion occupies 
the interval between stars of the 3d and 8th magnitudes. It is 
probable that some of these objects, which give no indications of a 
physical connection with stars, lie beyond the region of known 



STRUCTURE OF THE MATERIAL UNIVERSE. 229 

stars, but we have no means of obtaining even the remotest ap- 
proximation to the distance of individuals among them. 

606. A mere speck in the heavens, at the distance of the stars, 
as viewed through a good telescope, is as large as the earth's or- 
bit : accordingly the collections of nebulous matter which occur in 
the heavens, in the regions of the stars, must have, at least, as 
great a superficial extent as the orbit of the earth. Many of them 
must be vastly larger. For example, the nebula in Andromeda 
(599) is two-thirds of the apparent size of the moon's disc. Its 
actual extent cannot be less than 365,000 times that of the earth's 
orbit, or 1000 times that of the whole solar system. (See Note XX.) 

607. The matter of the smallest of these nebulae may be ex- 
ceedingly subtile, and yet be sufficient in quantity to condense into 
a body as large and as dense as the sun ; for it appears, by calcu- 
lation, that if the matter of the sun were to expand so as to fill the 
space enclosed within the earth's orbit, it would be about 45,000 
limes rarer than the air. 

STRUCTURE OF THE MATERIAL UNIVERSE— NEBULAR 
HYPOTHESIS. 

608. In view of the facts which have now been presented, it 
will be seen that the great prominent feature in the structure of the 
universe is the arrangement of the stars in detached beds. Thus 
our starry firmament is one immense bed of stars, in which occur 
a great number of subordinate clusters or beds, so that, in fact, it 
appears to be chiefly made up of more or less detached and con- 
densed groups of stars. Exterior to this stratum, as far as the 
telescope penetrates into the abyss of space, are seen other beds, 
apparently similar, for the most part, to those which occur within 
the stratum itself. But some that are seen, it is not improbable, 
are other firmaments constructed on the same vast scale as that of 
the milky way, and at a distance from it of 200 or 300 times its 
own diameter, (603.) Leaving these out of view, the others, al- 
though occurring here and there in almost every direction, beyond 
the stratum of the milky way, seem to be, the great majority of them, 
disposed in a stratum of unknown extent, crossing the stratum of the 
milky way nearly at right angles ; or rather, if the milky way was 
correctly gauged by Herschel, surrounding it, without anywhere 
touching it. These beds are of a great variety of forms. But the 
greater number of them are generally supposed to be spherical, or 
nearly so. Some which have been supposed to have this form, 
may, perhaps, be circular or elliptical strata, or of the form of 
spherical segments, more condensed towards the centre, and seen 
either perpendicularly in their true form, or obliquety, so as to have 
their longer axis foreshortened. Planetary nebulas may be such 
strata, which are not condensed towards the centre : and annular 
nebulae, the same, in which there is a deficiency of stars at the 
central parts. (See Note XXL) 



230 OF THE FIXED STARS 

609. The discoveries that have been made in the heavens seem 
then to point to this great truth, viz., that the plan upon which 
the universe has been fashioned, is that of an ascending scale of 
systems, from isolated suns with their attendant systems of planets, 
to the stupendous whole which fills the eye of the Infinite Creator. 

But the indications are that the work of creation is still in progress. IHspersed 
through the realms of space, as we have seen (p. 226), are immense masses of some 
sort of nebulous matter, which in their various stages of condensation upon one 
or more points or lines, seem to present in sibylline leaves the whole history of the 
progressive creation of existing worlds and systems of worlds, and at the same time 
to picture forth the accomplishment of a similar destiny on the part of these masses 
themselves. 

610. The theory that worlds have been and are still being 
slowly evolved from primordial nebulous masses by the gradual 
operation of the general forces and properties which the Creator 
has either permanently imparted to matter, or is incessantly 
renewing in it, is called the Nebular Hypothesis. Its author is 
Sir William Herschel. But Laplace, by undertaking to trace in 
detail the progress of the creation of the solar system, has still 
more effectually stamped his name upon it than the author him- 
self. The great arguments which are urged in its support are the 
following : 

(1.) That there is a multitude of shining nebulous masses now 
scattered throughout space, each of sufficient extent to furnish the 
materials of a world, and some perhaps of systems of worlds. 

(2.) That these masses present a long unbroken gradation, from 
a mass " without form and void" to a perfect star i that is, all the 
various states in which a single nebulous mass would be during 
the vast period that it occupies in condensing from its first rude 
formless state into a finished globe. 

(3.) That the universe, as it is, in both the general and particu- 
lar features of its structure, may be shown to be a natural me- 
chanical consequence of the hypothesis in question. 



PART III. 

OF THE THEORY OF UNIVERSAL GRAVITATION. 



CHAPTER XIX. 

OF THE PRINCIPLE OF UNIVERSAL GRAVITATION. 

611. It is demonstrated in treatises on Mechanics, that if a body 
move in a curve in such a manner that the areas traced by the 
radius-vector about a fixed point, increase proportionally to the 
times, it is solicited by an incessant force constantly directed to- 
wards this point. 

The following is a geometrical proof of this principle. Conceive the orbit to be 
a polygon of an infinite number of sides. Let ABCD (Fig. 113) be a portion of 
it ; and S the fixed point about which the radius- Yig. 113. 

vector describes areas proportional to the times, 
or equal areas in equal times. Since the impulses 
are only communicated at the angular points 

A, B, C, D, &c, of the polygon, the motion will j7 

be uniform along each of the sides AB, BC, CD, 
&c. : and since we may suppose the times of de- 
scribing these sides to be equal, we shall have the 
triangular area SAB equal to the triangular area 
SBC, and SBC equal to SCD, &c. Produce AB 
and make Be equal to AB, which may be taken to 
represeut the velocity along AB ; and join Cc. Cc 
will be parallel to the line of direction of the impulse 
that lakes effect at B. Upon SB let fall the per- 
pendiculars Am, en, Cr. Then, since AB = Be, 
Am = en ; and since the equivalent triangles 
SAB, SBC, have a common base SB, Am = Cr. 
It follows, therefore, that en = Cr, and conse- 
quently, that Cc is parallel to BS. The impulse 
which the body receives at B is therefore directed 

from B towards S. In the same manner it may ^ 

be shown that the impulse which it receives at G is directed from C towards S. 
The line of direction of the force passes, therefore, in every position of the body, 
through the point S. 

Now, by Kepler's first law, the areas described by the radii- 
vectores of the planets about the sun, are proportional to the times. 
It follows therefore from this law, that each planet is acted upon 
by a force which urges it continually towards' the sun. 

This fact is technically expressed by saying that the planets 
gravitate towards the sun, and the force which urges each planet 
towards the sun is called its Gravity y or Force of Gravity, towards 
the sun. 

612. It is also proved by the principles of Mechanics, that if a 
body, continually urged by a force directed to some point, describe 




232 



OF THE PRINCIPLE OF UNIVERSAL GRAVITATION, 



an ellipse of which that point is a focus, the force by which it is 
urged must vary inversely as the square of the distance. 

Fig 114. Thus, let ABG (Fig. 114) be the 

supposed elliptic orbit of the body, 
CA and CB its semi-axes, and S the 
focus towards which the force is 
constantly directed. Also let P be 
one position of the body, PR a tan- 
gent to the orbit at P ; and draw 
RQ parallel to PS, Quv, HI, and 
CD, parallel to PR, Qx perpendicu- 
lar to SP, PF perpendicular to CD, 
and join S and Q. CP and CD are 
semi-conjugate diameters. Denote 
them, respectively, by A' and B' ; 
and denote the semi-axes, CA and 
CB, by A and B. Since HI is par- 
allel to PR, and, by a well-known 
property of the ellipse, the angle RPS is equal to the angle HPT, PH is equal to 
PI : and since HC = SC, and CE is parallel to HI, E is the middle of SI. We 
have, therefore, 




PE = 



PS -f- PI PS 4- PH 



= CA = A. 



Now the force at P is measured by 2Pm ; and we may state the proportion 

A' 
Pu : Pu : : PE : PC : : A : A' ; which gives Tv = Pu — . 

By the equation of the ellipse referred to its centre and conjugate diameters, 
PG and DL, 

rv2 B'2 A' 

Q? =£72- (P» XG.)=^ (Pu — X Gv). 

If we regard Q as indefinitely near to P, then Qu = Q», and Gv = 2CP = 2A' ; 
and therefore 



Qu 



B2 „ A' 



B'2 
.2A')=^r.2Pu 
A 



(«0 



But Qu : Qx : : PE : PF : : C A : PF : 

and, by analytical geometry, 

CD X PF = C A X CB, or, CA : PF : : CD : CB r : B' : B. 



Hence Qu : Qx : : B' 



B, Qu 2 



Qa; : : B' 



\ andQ« 2 =Qx 2 |? 



Substituting in equation (a), 



— 9 B'2 B'2 

Q *V=r- 2P * 



— 2 B 2 

whence Qx = --— . 2Pu. 
A 



Qx — 2 4A-2 

Now triangular area SQP = k = SP X -5- ; whence Qx = z=. . Substituting, 



SP2 



there results 



4/<2 



B 2 A 

~.2Pa,or2Pu = £- 
A B2 



4R 



SP 2 



(I). 



To compare the intensities of the force at different points of the orbit, we must 
take the values of 2P«, by which they are measured, for the same interval of time. 
On this supposition k is constant, and therefore the force is inversely proportional 
to the square of the distance SP. 

It therefore follows from Kepler's second law, viz. : that the 
planets describe ellipses having the centre of the sun at one of 



233 

their foci ; that the force of gravity of each planet towards the 
sun varies inversely as the square of the distance from the sun's 
centre. 

613. By taking into view Kepler's third law, it is proved that it 
is one and the same force, modified only by distance from the sun, 
which causes all the planets to gravitate towards him, and retains 
them in their orbits. This force is conceived to be an attraction 
of the matter of the sun for the matter of the planets, and is called 
the Solai' Attraction. 

To deduce this consequence from Kepler's third law, let t, V, denote the periodic 
times of any two planets ; r, r', their distances from the sun at any assumed point 
of time ; k, k', the areas described by them in any supposed unit of time ; and A, B, 
and A', B', the semi-axes of their elliptic orbits. Then kt, k't, will be equal to the 
areas of the entire orbits ; which are also measured by 7rAB, 7tA'B'. 
Thus kt : k't' : : AB : A'B', and £2*2 : ftty? : : A2B2 : A'2B'2 

But, by Kepler's third law, t* . t >2 : : A3 : A'3. 

B2 B'2 

Dividing, and reducing, £ 2 : &' 2 ::—:-— : 

that is, the squares of the areas described in equal times are as the parameters of 
the orbits. 

Now, let /, /', denote the forces soliciting the two planets. Then, by equation 
(I), Art. 612, 

, A', ilB 'l . p A'2 1 

A B2_ J_ A/_ B2 J_ 

B2 ' A ' r2 : B'2* A' 't*' 







J ~ B2 ' ' r2 ' J 


whence 


/:/: 


• A k 2 LA &L. 

'B2 >2 "B'2 V'2 ' 


or 




f:f::-: -■ 



From which it appears that the planets are solicited by a force of gravitation 
towards the sun, which varies from one planet to another according to the law of 
the inverse square of their distance. 

614. The motions of the satellites are in conformity with Kep- 
ler's laws ; hence, the planets which have satellites are endued 
with an attractive force of the same nature with that of the sun. 

615. The existence of a similar attractive power in each of the 
planets that are devoid of satellites, is proved by the fact that the 
observed inequalities of their motions, and of those of the other 
planets, may be shown upon this supposition to be necessary con- 
sequences of the attractions of the planets for each other. 

616. In like manner the inequalities in the motions of the satel- 
lites and their primaries, show that the satellites possess the same 
property of attraction as the sun. 

617. We learn from the motions produced by the action of the 
sun and planets upon each other, that the intensities of their at- 
tractive forces are, at the same distance, proportional to their 
masses, and that the whole attraction of the same body for differ- 
ent bodies, is, at the same distance, proportional to the masses of 
these bodies. From which we may infer that a mutual attraction 
exists between the particles of bodies, and that the whole force of 
attraction of one body for another, is the result of the attractions 

30 



234 THEORY OF THE ELLIPTIC MOTION OF THE PLANETS. 

of its individual particles. Moreover, analysis shows, that in or- 
der that the law of attraction of the whole body may be that of the 
inverse ratio of the square of the distance, this must also be the 
law of attraction of the particles. The fact, as well as the law of 
the mutual attraction of particles, is also revealed by the tides and 
other phenomena referable to such attraction. 

618. The celestial phenomena compared with the general laws 
of motion, conduct us therefore to this great principle of nature ; 
namely, that all particles of matter mutually attract each other in 
the direct ratio of their masses, and in the inverse ratio of the 
squares of their distances. This is called the principle of Univer- 
sal Gravitation. The theory of its existence was first promul- 
gated by Sir Isaac Newton, and is hence often called Newton's 
Theory of Universal Gravitation. The force which urges the 
particles of matter towards each other is called the Force of Grav- 
itation, or the Attraction of Gravitation. 

619. In the following chapters our object will be to develope the 
most important effects of the principle of gravitation thus arrived at 
by induction. The perfect accordance that will be observed to 
obtain between the deductions from the theory of universal gravi- 
tation and the results of observation, will afford additional confir- 
mation of the truth of the theory. 



CHAPTER XX. 

THEORY OF THE ELLIPTIC MOTION OF THE PLANETS. 

620. Let the attraction of the unit of mass of the sun for the 
unit of mass of a planet, at the unit, of distance, be designated by 1. 
The whole attraction exerted by the sun upon the unit of mass, at 
the same distance, will then be expressed by the mass of the sun 
(M) ; or, in other words, by the number of units which its mass 
contains. And the attraction F, at any distance r, will result from 

M 

the proportion M : F : : r 2 : l 2 , which gives F= -^. This, in the lan- 
guage of Dynamics, is the Accelerating Force soliciting the planet. 

M 

As — expresses the attraction of the sun for a unit of mass of 

the planet, its attraction for the entire mass m of the planet will be 

M 

expressed by m — . This is the moving force of the planet, and 

since it is, at the same distance, pioportional to the mass of the 



REVOLUTION ABOUT THE CENTRE OF GRAVITY. 



235 



planet, the velocity due to its action is the same, whatever may be 
the mass. 

621. The planet has also an attraction for the sun, as well as the 
sun for the planet, and the expression for its attractive force, or for 

771 

the accelerating force animating the sun, will obviously be -g. The^ 

sun will then tend towards the planet, as the planet towards the 
sun. But, if the two bodies were to set out from a state of rest, 
the velocity of the planet would be as many times greater than the 
velocity of the sun, as the mass of the sun is greater than that of 
the planet. For the velocity of the planet would be to that of the 
sun as the attractive force of the sun is to the attractive force of 

tne planet, that is, as -^ : -§, or as M : m. 

As the attraction of the particles of the sun and planet are mu- 
tual and equal, the attraction of the planet for the entire mass of 
the sun must be equal to the attraction of the sun for the entire 
mass of the planet. 

622. The sun and any 'planet revolve about their common cen- 
tre of gravity. 

To show this, we would remark, in the first place, that it is a 
principle of Mechanics that the mutual actions of the different 
members of a system of bodies cannot affect the state of the centre 
of gravity of the system. This is called the Principle of the 
Presei'vation of the Centre of Gravity. It follows from it that 
the common centre of gravity of the sun and any planet is at rest, 
unless it has a motion of translation in common with the two bo- 
dies, imparted by a force extraneous to the system. As we are 
concerned at present only with the relative motion of the sun and 
planet, such motion of translation, if it does exist, may be left 
out of account. Now, let S (Fig. 115) be the Fig. 115. 

sun, and r any planet, supposed for the mo- 
ment to be at rest. If neither of the two bo- 
dies should receive a velocity in a direction 
oblique to PS, the line of their centres, they 
would move towards each other by virtue of 
their mutual attraction, and meet at C their 
common centre of gravity.* But, if the body 
P have a projectile velocity given to it in any 
direction Vt, inclined to the line PS, it is sus- 
ceptible of proof that its motion relative to the 
sun may be in an ellipse, as is observed to 
be the case with the planets. 

Now, while the planet moves in space, the line of the centres 




* The common centre of gravity of two bodies lies on the line joining their cen- 
tres, and divides this line into parts inversely proportional to the masses of the 
bodies. 



236 THEORY OF THE ELLIPTIC MOTION OF THE PLANETS. 

of the planet and sun must continually pass through the stationary- 
position of the centre of gravity ; and therefore, when the planet has 
advanced to any point p, the sun will have shifted its position to 
some point s on the line pC prolonged. Moreover, as the two bo- 
dies mutually gravitate towards each other, the paths of each in 
space will be continually concave towards the other body, and 
therefore also towards the centre of gravity C, which is constantly 
in the same direction as the other body. Since the planet per- 
forms a revolution around the sun, the sun and planet must each 
continue to move about the point C until they have accomplished 
a revolution and returned to the line PCS. Also, as the distance 
PS of the two bodies will be the same at the end as at the begin- 
ning of the revolution, as well as the ratio of their distances PC 
and SC from the centre of gravity, they will return to the posi- 
tions P, S, from which they set out, and will therefore move in 
continuous curves. 

Moreover, these curves are similar to the apparent orbit described by P around 
S. For, draw Sp' parallel and equal to sp, and join Yp and Ss. Then, since 
sC : Cp : : SC : CP, Fp is parallel to Ss ; and therefore Fp produced passes 
through p'. Whence, CP : Cp : : SP : Sp'. Moreover, the angle PC^ = FSp'. . It 
follows, therefore, that the area YOp is similar to the area YSp' ; and thus that the 
orbit of P around C is similar to the apparent orbit of P around S. The latter is 
known from observation to be an ellipse. The former is therefore also an ellipse. 

As the distances of the sun and planet from their common cen- 
tre of gravity are constantly reciprocally proportional to their 
masses, the orbit of the sun will be exceedingly small in compari- 
son with the orbit of the planet. 

623. If to both the sun and planet there should be applied a 

771 

force equal to the accelerating force of the sun, ~g, (621), but in an 

opposite direction, the sun would be solicited by two forces that 
would destroy each other, but the planet would now be urged 
towards the sun remaining stationary, with the accelerating force 

— ~2 — , or a force the intensity of which was equal to the sum of 

the intensities of the attractive forces of the sun and planet, at the 
distance of the planet. Now, the application of a common force 
will not alter the relative motion of the two bodies. Hence, in in- 
vestigating this motion, we are at liberty to conceive the sun to be 
stationary, if we suppose the planet to be solicited by the accelera- 
ting force — -^ — . As the mass of the sun is very much greater 

than that of any planet, but little error will be committed in neg- 
lecting the attraction of the planet, and taking into account only 

• M 

the sun s action —5. 

r 2 

624. Analysis makes known the general laws of the motion of 
a body, when impelled by a projectile force, and afterwards contin- 



GENERAL PROBLEM OF MOTION OF REVOLUTION. 237 

ually attracted towards the sun's centre by a force varying inverse- 
ly as the square of the distance. We learn by it that the body 
will necessarily describe some one of the conic sections around 
the sun situated at one of its foci. We learn, also, that the na- 
ture of the orbit, as well as the length of the major axis, is wholly 
dependent, for any given distance of the planet, upon the intensity 
of the projectile force, but that the position of the axis, and the ec- 
centricity of the orbit, depend also upon the angle of projection, 
(that is, the angle included, at the commencement of the motion, 
between the line of direction of the projectile force and the radius- 
vector.) As to the relative intensity of the projectile force neces- 
sary to the production of each one of the conic* sections, a certain 
intensity of force will produce a parabola ; any less intensity, an 
ellipse or circle ; and any greater, an hyperbola. 

625. If the velocity that would at a given distance be imparted 
by the sun's attraction in a second of time, which is the measure 
of its intensity at the given distance, be found, and also the dis- 
tance of a planet at any time, as well as its velocity and the angle 
made by the direction of its motion with the radius-vector, the form, 
dimensions, and position of the planet's orbit can be computed. 
This is to determine the orbit a priori. The practice has been, 
however, to determine the various elements of a planet's orbit by 
observation, (as already described, Chap. VII.) 

The elements being known, the equations of the elliptic mo- 
tion, investigated on the principles of Mechanics, serve to make 
known the position and velocity of the planet at any time. (The 
investigation of these equations may be found in the Encyclopaedia 
Metropolitana, Article Physical Astronomy, page 653, in the Me- 
canique Elementaire de Francoeur, and in many other similar 
works.)* 

626. The physical theory of the motion of a satellite around its 
primary is obviously the same as that of the motion of a planet 
around the sun. 

627. According to the principle of the preservation of the centre 
of gravity (622), the centre of gravity of the whole solar system 
must either be at rest, or have a motion of translation in space in 
common with the system, resulting from the action of a foreign 
force. We have already seen (593) that it has been ascertained 
from observation, that it is in fact in motion. 

628. The sun and planets revolve around their common centre 
of gravity. The path of the sun's centre results from the joint ac- 
tion of all the planets, and is a complicated curve. As the quan 
tity of matter in all the planets taken together is very small, com 
pared with that in the sun, (less than 1 ~ i ) the extent of the curve 
described by the centre of the sun cannot be very great. It is 

* The equations are the same with those deduced directly from Kepler's laws of 
Che planetary motions. 



238 THEORY OF THE ELLIPTIC MOTION OF THE PLANETS. 

found by computation, that the distance between the sun's centre 
and the centre of gravity of the system can never be equal to the 
sun's diameter. 

629. It is demonstrated in treatises on Mechanics, that if foreign 
forces act upon a system of bodies, the centre of gravity of the sys- 
tem will move just as the whole mass of the system concentrated 
at the centre of gravity would move, under the action of the same 
forces. It follows from this principle, that from the attraction of 
the sun for a primary planet and its satellites, their common cen- 
tre of gravity will revolve around the sun, just as the whole quan- 
tity of matter in the planet and its satellites concentrated at this 

'point would, under the influence of the same attraction. Moreover, 
the same considerations which show that the sun and planets re- 
volve about their common centre of gravity, will also show that a 
primary planet and its satellites revolve about their common centre 
of gravity. It appears, therefore, that in the case of a planet 
which has satellites, it is not, strictly speaking, the centre of the 
planet that moves agreeably to the first and second laws of Kepler, 
but the common centre of gravity of the planet and its satellites ; 
the planet and satellites revolving around the centre of gravity, as 
it describes its orbit about the sun. 

630. It may be worth while here to remark, that the revolution 
of the earth around the common centre of gravity of the earth and 
moon, occasions an inequality, both of longitude and latitude, in 
the apparent motion of the sun. It is, however, exceedingly small, 
for the reason that the distance of the earth's centre from the cen- 
tre of gravity is very short, in comparison with the distance of the 
sun. The mass of the earth is to that of the moon as 80 to 1, 
while the distance of the moon is to the radius of the earth as 60 
to 1 : it follows, therefore, that the common centre of gravity of the 
earth and moon lies within the body of the earth. 

631. It appears also from the physical investigation of the ellip- 
tic motion of the planets, that Kepler's third law is not rigorously 
true. In consequence of the action of the planets upon the sun, 
the ratio of the periodic times of the different planets depends upon 
the masses of the planets, as well as their distances from the sun. 
If p and p' be the periodic times of any two of the planets, a and a' 
their mean distances from the sun's centre, and m and m! their 
quantities of matter, that of the sun being denoted by 1, then, dis- 
regarding the actions of the other planets, 



As m and m' are very small fractions, the error resulting from their 
omission will be very small. If we omit them, we shall have 

p 2 :p' 2 : : a 3 : a' 3 ; 
which is Kepler's third law. 



INVESTIGATION OF THE DISTURBING FORCES. 



239 



CHAPTER XXI. 

THEORY OF THE PERTURBATIONS OF THE ELLIPTIC MOTION OF THE 
PLANETS AND OF THE MOON. 



632. We have, in a previous chapter, given a general idea of the mode of 
determining, from theory and observation combined, the law and amount of 
the perturbations or inequalities of the lunar and planetary motions. We pro- 
pose now to give some insight into the nature and manner of operation of the 
disturbing forces, and will commence with the perturbations of the moon pro- 
duced by the action of the sun. 

633. We have already (283) shown how the intensity and direction of 
the disturbing force of the sun, in any given position of the moon in its orbit, 
may be determined. Let us now derive the disturbing forces that take effect 
in the three directions in which the motion of the moon can be changed ; 



Fig. 116. 



namely, in the direction of the radius- 
vector, of the tangent to the orbit, and 
of the perpendicular to its plane. Let E 
(Fig. 116) be the earth, M the moon, and 
S the sun. Let the force exerted by the 
sun upon the moon be decomposed into two 
forces, one acting along the line MS' par- 
allel to ES, and the other from M towards 
E. If the component along MS' were equal 
to the force exerted by the sun upon the 
earth, the motion of the moon about the 
earth would not be changed by the action 
of these two forces. Hence, the difference 
between them will be the disturbing force in 
the direction MS'. The component along 
ME is another disturbing force. It is called 
the Addititious Force, because it tends to 
increase the gravity of the moon towards the 
earth. The disturbing force along MS' will 
generally be inclined to the plane of the / 
orbit, and may be decomposed into three N 
forces, one in the direction of the tangent, 
another in the direction of the radius-vec- 
tor, and a third in the direction of the per- 
pendicular to the plane. The first men- 
tioned component is called the Tangential 

Force ; the second is called the Ablatitious Force ; and the third we shall call 
the Perpendicular Force. 

The actual disturbing force in the direction of the radius-vector is equal to 
the difference between the addititious and ablatitious forces, and is called the 
Radial Force. This and the tangential and perpendicular forces constitute 
the disturbing forces, the direct operation of which is to be considered. 

63-1. To obtain general analytical expressions for these forces, let the dis- 
tance of the sun from the earth (which for the present we shall suppose to 
be constant) be denoted by <z, and the distances of the moon from the earth 
and sun, respectively, by y and z. Also let F = the force exerted by the 
earth upon the moon, P = the force exerted by the sun upon the earth, and 
Q = the force exerted by the sun upon the moon. Then, if we denote the 




240 



PERTURBATIONS OF ELLIPTIC MOTION OP THE MOON. 



mass of the earth by 1, and take m to stand for the mass of the sun, we shall 
have, (620,) 

z 3 

Let the force Q be represented by the line MS (Fig. 116) ; and let its 
component parallel to ES, or MS' = R, and its component along the radius- 
vector, or ME = T. 






Q : T : : MS : ME ; or, -J : T : : * : y. 



s? 



addititious force T = ^j . . . (130). 



Whence, 
In a similar manner we obtain 



(131). 

The disturbing force in the direction of the sun 



ma 



ma m / 1 1 \ 

= tt — r = — 3 — — =ma[ -=- - I . 

z* a 2 yz 3 a? / 



Now, let a, 0, y, denote the angles made by the line MS', respectively, with 
the tangent, the radius-vector, and the perpendicular to the plane of the orbit, 
and we shall have for the components of the disturbing force R — P, along 
these lines ; 



/_1_ 1 \ 

tangential force = ma I z a — ~~f ) cos a 

ablatitious force = ma( — | cos B 

\ z* a 3 ) 

a ("?"~^) C0S: 



perpendicular force 
Fig. 117. 



(132) ; 



(133) ; 



(134). 




Combining equation (133) with equation (130) 
we obtain for the radial force, 



radial force = my-j — ma 



(*-*) 



^T |COS0. 



635. The obliquity of the orbit of the moon 
to the plane of the ecliptic, affects but very 
slightly the value of the tangential and radial 
forces. If we leave it out of account, or sup- 
pose the moon's orbit to lie in the plane of the 
ecliptic, we shall have (Fig. 117) /?==S'ML 
= SEM the elongation of the moon = ^, and 
a = complement of 0, which gives 



tang, force = ma ( — — — I sin 
rad. force=my — ma I -- - J 



(135); 



cos <p (136). 



636. Equation (134) may be transformed 
into another, which is better adapted to the 
purposes we have in view. Let MK (Fig. 
116) represent the perpendicular to the plane 
of the moon's orbit, MF the intersection of the plane SMK with the plane 
of the moon's orbit, and SI, IF the intersections of a plane passing through 



INVESTIGATION OF THE DISTURBING FORCES. 241 

S and perpendicular to EN, the line of nodes, with the plane of the ecliptic 
and the plane of the orbit. SF will be perpendicular to both IF and MF. 
Denote SIF, the inclination of the orbit to the ecliptic, by I, SEN the angu- 
lar distance of the sun from the node by N, and SE and SM by a and z, as 
before. 

Now, in equation (134) y stands for the angle S'MK, but S'MK = SMK, 
(nearly,) and 

cos SMK = sin SMF == g^-. 

SF = SI sin SIF, and SI = SE sin SEI; 
whence SF = SE sin SEI sin SIF = a sin N sin I : 

substituting, 

a sin N sin I a sin N sin I 

cos y = cos SMK = — — = . 

SM z 

Thus we have 

/ 1 1 \ a sin N sin I 

perpen. force = ma I ■; — — 1 ■ . . . (137). 

\z a / z 

637. The variable z may be eliminated from equations (135), (136), and 
(137), and other equations obtained, involving only the variables y and <p. Let 
ML (Fig. 116) be drawn through the place of the moon perpendicular to ES. 
Then, using the same notation as in the preceding articles, 

LS = z (nearly), EL = EM cos LEM= y cos <p. 
But LS = SE — EL; 

whence z = a — y cos <p, and z 3 = a 3 — 3<z 2 y cos <p : 

neglecting the terms containing the higher powers of y than the first, as they 
are very minute, y being only about ^\^ a. 

1 1 1 3y cos <t> 

1? == a 3 — 3a 2 y cos <p == ~^ ^ ' 

neglecting all the terms of the quotient that involve higher powers of y than 

the first. Substituting this value of — ■ in equation (135), we obtain, 

. , /. 3my cos a sin & 
tangential force = ^ ; 

or, (App. For. 13), 

3my sin 2<p 
tangential force = -— — ■ . . . (138). 

2 CL 

Making the same substitution in equation (136), and neglecting the term con- 
taining y 2 , there results, 

my (1 — 3 cos 2 0) 
. radial force = ^ ; 

or, (App. For. 9), 

,. , Tny (l-f-3cos2<*) 
radial force = — ^-^ J b — ^ . . . (139). 

In equation (137) we have to substitute, besides, the value of 2, viz. a — y 
cos ^ ; then dividing and neglecting as before, we have 

3wycos0 . ^ 
perpen. force = - 3 sin N sin I . . . (140.) 

638. If the disturbing forces retained constantly the same intensity and di- 
rection, the result would be a continual progressive departure from the ellip- 
tic place ; but, in point of fact, these forces are subject to periodical changes 
of intensity and direction from several causes, from which results a compen- 

31 



242 PERTURBATIONS OF ELLIPTIC MOTION OF THE MOON. 

sation of effects, and an eventual return to the elliptic place. The causes of 
the variation of the disturbing forces are : 

(1.) The revolution of the moon around the earth. 

(2.) The elliptic form of the apparent orbit of the sun. 

(3.) The elliptic form of the orbit of the moon. 

(4.) The inclination of the two orbits. 

As the variations of the radial and tangential forces, resulting from the in- 
clination of the orbits, are very minute, we shall leave them out of account, 
and in the consideration of the effects of these forces shall, for the sake of 
simplicity, regard the orbits as lying in the same plane. 

The first mentioned circumstance is the most prominent cause of variation, 
and gives rise to the more conspicuous perturbations. The other two serve 
to modify the variations of the forces resulting from the first, and occasion 
each a distinct set of periodical perturbations. 

639. Let us now investigate, in succession, the effects of eacn of the dis- 
turbing forces, commencing with the tangential force. The tangential force 
takes effect directly upon the velocity of the moon in its orbit ; and as its line 
of direction does not pass through the earth, it disturbs the equable descrip- 
tion of areas. It also affects the radius-vector 1 indirectly, by changing the 
centrifugal force. To understand the detail of its action we must inquire in- 
to the variations which it undergoes. 

If we regard y as constant in the expression for the tangential force, (equa. 
138,) which amounts to considering the moon's orbit as circular, the expres- 
sion will become equal to zero when sin 2<p = 0, and will have its maximum 
value when sin 20= 1. It will also change its sign with sin 20. It appears, 
therefore, that the tangential force is zero in the syzigies and quadratures, 
where it also changes its direction, and that it attains its maximum value in 
p. , ,g the octants. It will be seen, on inspect 

ing Fig. 118, that it will be a retarding 
force in the first quadrant, (AB). Accord- 
ingly, it will be an accelerating force in 
the second, a retarding force again in the 
third, and an accelerating force again in 
the fourth. 

This will also appear upon considering 
the direction of the disturbing force par- 
allel to the line of the centres of the sun 
and earth, in the various quadrants. In 
the nearer half of the orbit the sun tends 
to draw the moon away from the earth, 
and the force in question is directed to- 
wards the sun. In the more remote half 
the sun tends to draw the earth away from the moon, but we may regard it, 
instead, as urging the moon from the earth by the same force ; for the rela- 
tive motion will be the same on this supposition. In the part of the orbit 
supposed, then, the disturbing force under consideration will be directed from 
the sun, as represented in Fig. 118. 

640. It appears, then, that the tangential force will alternately retard and 
accelerate the motion of the moon during its passage through the different 
quadrants, and that the maximum of velocity will occur in the syzigies, A, C, 
where the accelerating force becomes zero, and the minimum of velocity in 
the quadratures, B, D, where the retarding force becomes zero. On the sup- 
position that the orbit is a circle, the arcs AB, BC, CD, and DA, would be 
tfqual, and the retardation of the velocity in one quadrant would be compen- 
sated for by an equal acceleration in the next, and at the close of a synodic 
revolution the velocity of the moon would be the same as at its commence- 
ment. As the velocity is greatest in the syzigies and least in the quadratures, 
and as the degree of retardation is the same as that of acceleration, the medn 




EFFECTS OF THE TANGENTIAL FORCE. 24S 

motion* must have place in the octants. Now, as the moon moves from the 
syzigy A with a motion greater than the mean motion, her true place will be 
in advance of her mean place, and will become more and more so till she 
reaches the octant, where the true motion is equal to the mean. The dif- 
ference between the true and mean place will then be the greatest ; for after 
that, the true motion becoming less than the mean, the mean place will ap- 
proach nearer to the true, till at the quadrature they coincide. Beyond B, 
the true motion still continuing less than the mean, the mean place will be in 
advance of the true, and the separation will increase till at the octant the 
true motion has attained to an equality with the mean motion, after which, the 
mean motion being the slowest, the true place will approach the mean till at 
the syzigy C they again coincide. Corresponding effects will take place in 
the two remaining quadrants. We perceive, therefore, that the tangential 
force produces aR inequality, of longitude, which attains to its maximum posi- 
tive and negative value in the octants, and is zero in the syzigies. This is the 
inequality known in Plane Astronomy by the name of Variation, (296.) 

641. Let us now inquire into the modifications of the effects of the tangen- 
tial force, that result from the elliptic form of the sun's orbit. Suppose that 
at the moment when the moon sets out from conjunction the sun is in the 
apogee of its orbit : then it is plain that, during the whole revolution of the 
moon, the sun's disturbing force would be on the increase by reason of the 
diminution of the sun's distance, and that, in consequence, the retardation in 
the first quadrant would be less than the acceleration in the second, and the 
retardation in the third less than the acceleration in the fourth. So that, 
when the moon had again come round into conjunction, the acceleration would 
have over-compensated the retardation. This kind of action would go on so 
Jong as the sun approached the earth ; but when it had passed the perigee of 
its orbit, and began to recede from the earth, the reverse effect would take 
place, and a retardation of the moon's orbitual motion would happen each 
revolution. If the anomalistic revolution of the sun was an exact multiple of 
the synodic revolution of the moon, the acceleration in each revolution of the 
moon during the passage of the sun from the apogee to the perigee of its or- 
bit, would be compensated for by an equivalent retardation in the revolution 
of the moon answering to the same distance of the sun in its passage from the 
apogee to the perigee ; and the velocity of the moon would be the same at 
the close of an anomalistic revolution of the sun as at its commencement. But 
as this relation does not, in fact, subsist between the anomalistic revolution 
of the sun and the synodic revolution of the moon, a compensation between 
the accelerations and retardations, answering to the different revolutions of 
the moon, will not be effected until conjunctions shall have occurred at every 
variety of distance of the sun in each half of its orbit. Since the anomalistic 
and synodic revolutions are incommensurable, the sun will be, in reality, in 
every variety of position in its orbit at the time of conjunction, in process of 
time ; so that eventually the original velocity in conjunction will be regained. 
It appears, therefore, that the variation of the moon's motion from one revo- 
lution to another, occasioned by the elliptic form of the sun's orbit, is periodic. 
Its period will be the interval of time in which the moon will perform a cer- 
tain number of synodic revolutions, while the sun performs a certain number 
of anomalistic revolutions. Avoiding unnecessary precision, we find it to con- 
sist of but a moderate number of years. 

642. We have next to consider the consequences of the elliptic form of 
the moon's orbit. We remark, in the first place, that, the orbit being an 
ellipse, the areas AEB, BEC, CED, and DEA, (Fig. 118,) will be unequal, 
and therefore, by the laws of elliptic motion, the arcs AB, BC, CD, and DA, 
will be described in unequal times. It follows from this, that the retardation 

* The expressions, mean motion, true motion, mean place, true place, are here 
Ito be understood only in relation to the perturbation under consideration. 



244 PERTURBATIONS OF ELLIPTIC MOTION OF THE MOON 

in the first quadrant will not be exactly compensated by the acceleration nr 
the second, and that the retardation in the third will not be exactly compen- 
sated by the acceleration in the fourth. Therefore, at the end of the synodic 
revolution the moon will hare an excess or deficiency of velocity. Its mean 
motion will then vary from one revolution to another,, by reason ©f the ellip- 
ticity of its orbit. This variation will be periodic, like that just considered,, 
and for similar reasons. The excess or deficiency of velocity at the close of 
any one revolution^ will in time be compensated by an equal efficiency or 
excess occurring at the close of another revolution, when the sun has a cer- 
tain different position with respect to the perigee of the moon's orbit. 

643. We pass now to the consideration of the action of the radial force. 
The direct general effect of the radial force, is an alteration in Ihe intensity 
of the moon's gravity towards the earth, and in its law of variation. Its 
specific effects are periodical variations in the magnitude, eccentricity, and 
position of the orbit. As it is directed towards the earth, it will not disturb 
the equable description of areas. To discover the variations of this force 
we have only to discuss the general analytical expression for it T already in- 
vestigated. It is, 

my (I — 3 cos 2 f) 

radial force = ; . 

a 3 

We shall have radial force = 0, when 1 — 3 cos 2 p = (X r or when cos 
<p = ± </!• This value of cos ^ answers to four points lying on either side 
of the quadratures, and about 35° distant from them. When cos * is numeri- 
cally greater than */\ the result will be negative, and when it is less than 
y| the result will be positive. It follows, therefore, that the radial force- 
increases the gravity of the moon in the quadratures, and for about 35° on 
each side of them y and that during the remainder of a synodie revolution ife 
diminishes it. 

When the moon is in quadratures, cos f = 0, and 

radial force = ^y . . . . (141). 
In the syzigies, we have cos £ = ± 1, which gives 
radial force =~-^r . , . (142), 

It appears, then, that the diminution of the moon's- gravity in the syzigies 
is double of its increase in the quadratures. 

We learn also from equations (141) and (142), that the radial force in the 
quadratures and syzigies varies directly as the distance ; from which we con- 
clude that the gravity of the moon varies at these points by a different law 
from that of the inverse squares. In the quadratures the gravity will be in- 
creased most at the greatest distance, where it is the least ; and thus it will; 
vary in a less rapid ratio than the square of the distance. In the syzigies it 
will be diminished most at the greatest distance, or where it is the least ; and 
accordingly, at these points it will vary in a more rapid ratio than the square; 
of the distance. 

644, An easy investigation, with the aid of the differential calculus, proves 

that the mean diminution of the moon's gravity from the sun's action is —^ " T 

r representing in this case the mean distance of the moon from the earth. 
The value of this expression is readily found to be equal to about the 360tb 
part of the whole gravity of the moon to the earth. 

In consequence of this diminution, the moon must describe her ©rbit at a 
greater distance from the earth, with a less angular velocity, and in a Iongei 
time, than if she were acted on only by the attraction of the earth. 

64&. The radial force of the sun alters the eccentricity of the moon's orbit. 



EFFECTS OF THE RADIAL FORCE. 245 

and differently in different revolutions of the moon, according to the position 
of the line of syzigies with respect to the line of apsides. When these lines 
are coincident the eccentricity is increased. ™ ,,0 

For, suppose PMAN (Fig. 119) to be the s ' 

-elliptic orbit of the moon that would be ^ — -— jyi 

described under the influence of a force y< ^v 

varying inversely as the square of the dis- / \ 

tance. In going from the apogee to the / \ 

perigee, the gravity will increase in a .[ e P u 

greater ratio than that of the inverse V " ~fi 

square of the distance ; the true orbit will \\ / / 

therefore fall within the ellipse, and tke \S^ s / 

.perigeasi distance (EP') will be less than ^^o~ -~^\s 

for the ellipse- Consequently, the eccen- ^--~__ __Z^~-<^ 

tricity will increase so much the more as 

the major axis diminishes. On the other hand, in going from the perigee to 
the apogee, the gravity will decrease in a greater ratio than the inverse square 
of the distance, and the mooa will consequently recede farther from the earth 
than if the orbit described was an ellipse. Therefore, in this half of the or- 
bit the eccentricity will also be increased. When the apsides are in quadra- 
tures the eccentricity will be diminished ; for the gravity will then vary from 
the apogee to the perigee, and from the perigee to the apogee, in a less ratio 
than that of the inverse squares ; and therefore the results will be contrary 
to those just obtained. The .eccentricity will have its maximum value when 
the apsides are in syzigies, and its minimum when they are in quadratures ; 
for, in every other position of the line of apsides with respect to the line of 
syzigies, the radial force in the apogee and perigee will be less tfean in these 
positions, (equa. 139.) and therefore alter less the proportional gravity of the 
moon in the apogee and perigee. It is evident, from the gradual decrease ot 
the radial force as we recede from the syzigies and quadratures, that the ec- 
centricity will continually diminish in the progress of the apsides from the 
-syzigies to the quadratures, and that it will continually increase from the 
quadratures to the syzigies. 

The change in the eccentricity of the moon's orbit, thus produced, will be 
attended with a corresponding change in the equation of the centre, and thus 
of the longitude. And this change is the conspicuous inequality of the moon, 
known by the name of Evectien, (296.) 

646. The radial force also produces a motion of the line of apsides. If the 
moon was only acted upon by the attraction of the earth its orbit would be an 
<ellipse, and the motion from one apsis to another, or, in other words, from 
one point where the orbit cuts the radius-vector at right angles to the other, 
would be 180°. In point of faet, however, the gravity due to the earth's 
attraction is constantly either diminished or increased by the radial disturbing 
force of the sun, and therefore its true orbit must continually deviate from 
the ellipse that would be described under the sole action of the earth's attrac- 
tion. When from the action of this force there is a diminution of the moon's 
gravity, she will continually recede from the ellipse in question, her path will 
be less bent, and she must therefore move through a greater angular distance 
before the central force will have deflected her course into a direction at right 
angles to the radius-vector. Accordingly, she will move through a greater 
angular distance than 180° in going from one apsis to another, and thus the 
apsides will advance. On the other hand, when the same force increases the 
moon's gravity, her path will fall within the ellipse, its curvature will be in- 
creased, and therefore it will be brought to intersect the radius-vector at right 
angles at a less angular distance. In this case, therefore, the apsides will 
move backward. Now, we have shown (643) that the radial disturbing force 
of the sun alternately diminishes and increases the moon's gravity to the earth. 
It follows, therefore, that the motion of the apsides will be alternately direct 



246 



PERTURBATIONS OF ELLIPTIC MOTION OF THE MOON. 



and retrograde ; but since, as has been shown, (643,) the diminution subsists 
during a longer part of the moon's revolution, and is moreover greater than 
the increase, the direct motion will exceed the retrograde, and therefore in 
an entire revolution the apsides will advance. 

647. The observed motion of the apsides of the moon's orbit is not, how- 
ever, wholly produced by the radial disturbing force. It is in part due to the 
action of the tangential force. This force alters the centrifugal force of the 
moon, and thus changes its gravity towards the earth, at the same time with 
the radial force. 

648. The elliptic form of the sun's orbit is the occasion of a change in. 
the radial force, from which results a perturbation of longitude called the An- 
nual Equation, (296.) The mean diminution of the moon's gravity, arising 

from the action of the sun, or the mean radial force,, is equal to — , (644.) 

Hence this diminution is inversely proportional to the cube of the sun's dis- 
tance from the earth. Therefore, as the sun approaches the perigee of its 
orbit, its distance from the earth diminishing, the mean diminution of the 
moon's gravity to the earth will increase,, and consequently the moon's dis- 
tance from the earth will become greater, and its motion slower, than it other- 
wise would be. The contrary will take place while the sun is moving from 
the perigee to the apogee. 

649. The disturbing force perpendicular to the plane of the moon's orbit, 
produces a tendency in the moon to quit that plane, from which there results 
a change in the position of the line of the nodes, and a change in the inclina- 
tion of the plane of the orbit to that of the ecliptic. If we examine the gene- 
ral expression for this force, viz : 

3/ny cos <l> . 
perpen. force = 3 sin JN sm I, 

we see that for any given values of N and I, it will be zero in the quadra- 
tures, and have its greatest value in the syzigies ; and that it will change its 
direction in the quadratures, lying, in the nearer half of the orbit, on the 
same side of its plane as the sun, and in the more remote half, on the opposite 
side. We perceive also that it will be zero for every value of £, or for every 
elongation of the moon, when the angle N is zero, that is,, when the sun is in 
the plane of the orbit ; and will attain its maximum, for any given elongation, 
when the line of direction of the sun is perpendicular to the line of nodes. 
It will also be the less, other things being the same, the smaller is the incli- 
nation I. 

650. Now let NM'R (Fig. 120) repre- 
sent the orbit of the moon, and S the sun, 
supposed stationary, the line of the nodes 
being in quadratures ; and let L, L' be the 
points of the orbit 90° distant from the 
nodes. The direction of the force, in the 
various points of the orbit, is indicated by 
S the arrows drawn in the figure. When the 
moon is at any point M' between L and the 
descending node N', she will be drawn out 
of the plane in which she is moving by the 
disturbing force M'K', and compelled to 
move in such a line as M7'. The node N' 
will therefore retrograde to some point n'. 
When she is at any point M, moving from 
the ascending node N towards L, her course 
will be changed to the line Mi, lying, like 
the line Wt', below the orbit, which being produced backward, meets the 
plane of the ecliptic in some point n, behind N. The nodes,, therefore,. retro- 



Fig. 120. 




EFFECTS OF THE PERPENDICULAR FORCE. 



247 



Fig. 121 



grade in this position of the moon, as well as in the former. When the moon 
is in 1 lie half N'L'N of the orbit, lying below the ecliptic, the absolute direc- 
tion of the disturbing force will be reversed, and thus its tendency will be the 
same as before, namely, to draw the moon towards the ecliptic. It follows, 
therefore, that throughout this half of the orbit, as in the other, the motion of 
the nodes will be retrograde. Accordingly, when the nodes are in quadra- 
tures, or 90° distant from the sun, they will retrograde during every part of 
the revolution of the moon. 

651. Suppose the sun now to be fixed on the line of nodes, or the nodes to 
be in syzigies. In this case the perpendicular force* will be zero, (649,) and 
therefore there will be no disturbance of the plane of the moon's orbit. 

652. Next, let the situation of the sun be intermediate between the two 
just considered, as represented in Figs. 120 and 121. The effect of the dis 
turbing force will be the same as in the first situation from the quadrature q 
(Fig. 120) to the node N', and from the quadrature q' to the node N. But 
throughout the arcs X(/, X~y, the direction of the force, and therefore the 
effects, will be reversed. The node will then retrograde, as before, while the 
moon moves over the arcs qlS' and </'X, and advance while she is in the arcs 
Ny, Xy. But as the force is greatest over the arcs ^N', q'lS, which con- 
tain the syzigies, {649,) and as these arcs are also longer than the arcs ISq, 
Ny, the node will, on the whole, retrograde each revolution. The velocity 
of retrogradation will, however, be less than when the nodes are in quadra- 
tuxes, and proportionably less as the distance of the sun from this position is 
greater. 

In the position represented in Fig. 121, 
a direct motion will take place over the 
arcs q'^S' and <?N ; but as ]Sy and Ny the 
arcs of retrograde motion, are of greater 
extent than ^'X' and ^N, and moreover 
contain the syzigies, the retrograde motion 
in each revolution must exceed the direct, 
as before. 

If we suppose the sun to be situated on 
the other side of the line of nodes, the 
effect of the disturbing force will obviously 
be the same in any one position of the sun, 
as in the position diametrically opposite to 
it. It appears, then, that the line of the 
nodes has a retrograde motion in every 
possible position of the sun. 

653. We have thus far supposed the sun 
to remain stationary in the various posi- 
tions in which we have supposed it, during the revolution of the moon. It 
remains, then, to consider the effect of the sun's motion in this interval. And 
first, it is plain, that, as the sun advances from S towards X', (Fig. 120,) the 
arcs X^, Xy will increase, and the arcs ^X' and q'~S dimmish ; from which 
it appears, that, during the advance of the sun from the point 90° behind the 
descending node to this node, its motion in the course of each revolution of 
the moon will cause the retrograde motion of the node to be slower than it 
otherwise would be. While the sun moves from the ascending node to the 
90° from it, the effect of its motion will obviously be just the reverse of this. 
During its passage from the descending to the ascending node, the effect will 
be the same in either quadrant as in that diametrically opposite. 

The variation in the intensity of the perpendicular force conspires with 
the difference of situation of the sun and its motion during a revolution of the 
moon in diminishing or increasing, as the case may be, the velocity of retro-' 
gradation of the nodes. 

654. Let us now r treat of the change of the inclination of the orbit, results 




248 



PERTURBATIONS OF ELLIPTIC MOTION OP THE MOON. 



ing from the disturbing action of the sun. And first, if we refer to Fig. 120 
we shall see that when the nodes are in quadrature the inclination will dimin- 
ish while the moon is moving from the ascending node N to the point L 90° 
distant from it, and increase while she is moving from L to the other node 
N'. In the other half of the orbit the tendency of the disturbing force is the 
same, (650 ;) and therefore while the moon is moving from N' to L' the in- 
clination will diminish, and while she is moving from L' to N it will increase. 
The diminutions and increments will compensate each other, and the original 
inclination will be regained at the close of the revolution. 

When the nodes are in syzigies there will be no change of inclination, 
(649.) 

655. In the situations of the sun represented in Figs. 120 and 121 the 
inclination will decrease from q to L and from q' to L', and increase from L 
to q' and from L' to q, the effects being the same as when the nodes are in 
quadratures over the arcs ^L and LN' in Fig. 120, and NL and Lq' in Fig. 



Fig. 120-. 



Fig. 121. 




N' . *1 




121, and being reversed over the arcs Nq and Ny in Fig. 120, and qN and 
g'N' in Fig. 121. When the sun has the position represented in Fig. 120. 
the arcs of increase Ly' and Uq will be greater than the arcs of diminution 
qh and q'U. The disturbing force will also be greater in the former arcs 
than in the latter. In the position supposed, therefore, there will be, on the 
whole, an increase of inclination every revolution. When the sun is in the 
position represented in Fig. 121, the arcs of diminution qL and q'U will be 
the greater ; and the force in them will also be the greater. In this case, 
therefore, there will be a diminution of the inclination each revolution of the 
moon. 

When the sun is on the other side of the line of nodes, the results will be 
the same as in the positions diametrically opposite. 

656. To inquire now into the consequences of the sun's motion during the 
revolution of the moon. As the sun moves from S towards N' (Fig. 120) 
the arcs L^', Jj'q, over which there is an increase of the inclination, will in- 
crease ; and the arcs qJu, y'L', over which there is a diminution, will diminish. 
The motion of the sun will, therefore, in approaching the descending node, 
render the increase of the inclination each revolution of the moon greater than 
it otherwise would be. When the sun is receding from the ascending node, 
the corresponding arcs will experience corresponding changes, and therefore 
the diminution will now be less than if the sun were stationary. 

The results will be similar for the opposite quadrants on the other side of 
the line of nodes. 

•657. Since the inclination diminishes as the sun recedes from either node, 



PLANETARY PERTURBATIONS. 249 

and increases as it approaches either node, it will be the least when the nodes 
are in quadratures, and the greatest when they are in syzigies. 

It is important to observe that the change of inclination which we have 
been considering is modified by the retrograde motion of the node ; and thus, 
that, besides the variations of this element connected with the motions of the 
moon and sun, there is another extending through the period employed by the 
node in completing a revolution with respect to both the sun and moon. 

658. The perturbations of the elliptic motion of the moon, comprising ine- 
qualities of orbit longitude, and variations in the form and position of the orbit, 
which have now been under consideration, depend upon the configurations of 
the sun and moon, with respect to each other, the perigee of each orbit, and 
the node of the moon's orbit. Their effects will disappear when the configu- 
rations upon which they depend become the same. They are therefore pe- 
riodical. 

659. The perturbations of the motions of a planet, produced by the action 
of another planet, are precisely analogous to the perturbations of the motions 
of the moon, produced by the action of the sun. The disturbing forces are 
obviously of the same kind, and they are subject to variations from precisely 
similar causes. But, owing to the smallness of the masses of the planets and 
their great distances, their disturbing forces are much more minute than the 
disturbing force of the sun. From this cause, together with the slow rela- 
tive motion of the disturbing and disturbed body, the motion of the apsides and 
nodes, and the accompanying variations of eccentricity and inclination, are 
very much more gradual in the case of the planets than in the case of the 
moon. Their periods comprise many thousands of years, and on this account 
they are called Secular Motions or Variations. In consequence of the greater 
feebleness of the disturbing forces, the periodical inequalities are also much 
less in amount. Moreover, as the motion of a planet is much slower than 
that of the moon, and as the variations of its orbit are more gradual than 
those of the lunar orbit, the compensations produced by a change of configu- 
rations are much more slowly effected, and thus the periods of the inequali- 
ties are much longer. 

660. The motions of the moon would be subject to no secular variations if 
the apparent orbit of the sun were unchangeable ; but the secular variation 
of the eccentricity of the sun's orbit, which answers to an equal variation of 
the eccentricity of the earth's orbit, that is produced by the action of the 
planets, gives rise to a secular inequality in the motion of the moon, called 
the Acceleration of the Moon. This inequality was discovered from observa- 
tion. Its physical cause was first made known by Laplace. 



CHAPTER XXII. 



OF THE RELATIVE MASSES AND DENSITIES OF THE SUN, MOON, AND 
PLANETS , AND OF THE RELATIVE INTENSITY OF THE GRAVITY 
AT THEIR SURFACE. 

661. The perturbations which a planet produces in the motions 
of the other planets, depend for their amount chiefly upon the ra- 
tio of the mass of the planet to the mass of the sun, and the ratio 
of the distance of the planet from the sun to the distance of the 
planet disturbed from the same body. Now, the ratio of the dis- 

32 



250 RELATIVE MASSES OF THE SUN, MOON, AND PLANETS. 

tances is known by the methods of Plane Astronomy ; conse- 
quently, the observed amount of the perturbations ought to make 
known the ratio of the masses, the only unknown element upon 
which it depends. 

This is one method of determining the masses of the planets. 
The masses of those planets which have satellites may be found 
by another and simpler method, viz. : by comparing the attractive 
force of the planet for either one of its satellites with the attract- 
ive force of the sun for- the planet. These forces are to each other 
directly as the masses of the planet and sun, and inversely as the 
squares of the distances of the satellite from the primary and of 
the primary from the sun. Thus, calling the forces f } F, the 
masses ?n, M, and the distances d, D, we have 

J ' r * ' (? ' D 2 ' 

whence we obtain m : M : :fcP : FD 2 . If we regard the orbits as 
circles, then d and D will be the mean distances, respectively, of 
the satellite from the primary, and of the primary from the sun, 
and are given in tables II, III, and VI. The ratio of / to F is 
equal to the ratio of the versed sines of the arcs described by the 
satellite and primary, in some short interval of time ;* since these 
are sensibly equal to the distances that the two bodies are deflect- 
ed in this interval from the tangents to their orbits, towards the 
centres about which they are. revolving : and since the rates of 
motion and dimensions of the orbits of the planet and satellites are 
known, these arcs and their versed sines are easily determined. 

662. The second column of Table IV exhibits the relative 
masses of the sun, moon, and planets, according to the most re- 
ceived determinations, that of the sun being denoted by 1 . 

663. The quantities of matter of the sun, moon, and planets, as 
well as their bulks, being known, their densities may be easily 
computed ; for, the densities of bodies are proportional to their 
quantities of matter divided by. their bulks. The third column of 
Table IV contains the densities of the sun, moon, and planets, that 
of the earth being denoted by 1. It will be seen on inspecting it, 
that, for the most part, the densities of the planets decrease as we 
recede from the sun. 

664. The relative intensity of the gravity at the surface of the 
sun, moon, and planets, may also readily be found, when the 
masses and bulks of these bodies are known. For supposing 
them to be spherical, and not to rotate on their axes, the gravity 
at their surface will be directly as their masses and inversely as 
the squares of their radii, or, in other words, proportional to their 
masses divided by the squares of their radii. The centrifugal 
force at the surface of a planet, generated by its rotation on its 

* It is to be observed that the versed sines here mentioned relate to the actual 
arcs described in the two unequal orbits. 



EXPLANATION OF SPHEROIDAL FORM OF THE EARTH. 251 

axis, diminishes the gravity due to the attraction of the matter of 
the planet. The diminution thus produced on any of the planets 
is not, however, very considerable. The method of determining 
the centrifugal force at the surface of a body in rotation, is given in 
treatises on Mechanics. (See Courtenay's Mechanics, pages 250 
and 251.) 

The fourth column of Table IV exhibits the relative intensity 
of the gravity at the surface of the sun, moon, and planets, that at 
the surface of the earth being denoted by 1 . 



CHAPTER XXIII. 



OF THE FIGURE AND ROTATION OF THE EARTH ; AND OF THE PRE- 
CESSION OF THE EQUINOXES AND NUTATION. 

665. We have already seen (159) that measurements made upon 
the earth's surface establish that the figure of the earth is that of 
an oblate spheroid, and that the oblateness at the poles is about 3-^5 . 

666. From the amount and law of the variation of the force of 
gravity upon the earth's surface, ascertained by observations upon 
the length of the seconds' pendulum, it is proved that the matter 
of the earth is not homogeneous, but denser towards the centre, 
and that it is arranged in concentric strata of nearly an elliptical 
form and uniform density. 

The fact of the greater density of the earth towards its centre 
has also been established by observations upon the deviation of a 
plumb-line from the vertical, produced by the attraction of a moun- 
tain ; — the amount of the deviation being ascertained by observing 
the difference in the zenith distance of the same star, as measured 
with a zenith-sector on opposite sides of the mountain. To the 
north of the mountain the plummet was drawn towards the south 
and the zenith distance of a star to the north of the zenith was 
diminished ; while to the south of the mountain the plummet was 
drawn towards the north, and the zenith distance of the same star 
was increased by an equal amount : and thus the difference of the 
two measured zenith distances was equal to twice the deviation of 
f.he plumb-line from the true vertical in either of the positions of 
the instrument ; (allowance being made for the difference of lati- 
tude of the two stations, as determined from the distance between 
them and the known length of a degree.) 

Such observations were made for the purpose of determining the 
mean density of the earth by Dr. Maskelyne, in 1774, on the sides 
of the mountain Schehallien in Scotland. The observed deviation 
of the plumb-line made known the ratio of the attraction of the 
mountain to that of the whole earth, and thus the relative quanti- 
ties of matter in the mountain and earth. These being ascertained 



252 OF THE FIGURE AND ROTATION OF THE EARTH, ETC. 

and the figure and bulk of the mountain having been determined 
by a survey, the relative density of the earth and mountain became 
known by the principle mentioned in Art. 663, and thence the ac- 
tual density of the earth, the density of the mountain having been 
found by experiment. The result was, that the mean density of 
the earth is 4.95, the density of water being 1 . 

667. The spheroidal form of the surface of the earth and of its 
internal strata is easily accounted for, if we suppose the earth to 
have been originally in a fluid state. The tendency of the mutual 
attraction of its particles would be to give it a spherical form ; but 
by virtue of its rotation, all its particles, except those lying imme- 
diately on the axis, would be animated by a centrifugal force in- 
creasing with their distance from the axis. If, therefore, we con- 
ceive of two columns of fluid extending to the earth's centre, one 
from near the equator, and the other from near either pole, the 
weight of the former would by reason of the centrifugal force be 
less than that of the latter. In order, then, that they may sustain 
each other in equilibrio, that near the equator must increase in 
length, and that near the pole diminish. As this would be true at 
the same time for every pair of columns situated as we have sup- 
posed, the surface of the whole body of fluid about the poles must 
fall, and that of the fluid about the equator rise. In this manner 
the earth would become flattened at the poles and protuberant at 
the equator. 

668. Upon a strict investigation it appears that a homogeneous 
fluid of the same mean density, with the earth, and rotating on its 
axis at the same rate that the earth does, would be in equilibrium, 
if it had the figure of an oblate spheroid, of which the axis was 
to the equatorial diameter as 229 to 230, or of which the oblate- 
ness was ^io- If tne A m( i mass supposed to rotate on its axis be 
not homogeneous, but be composed of strata that increase in den- 
sity from the surface to the centre, the solid of equilibrium will 
still be an elliptic spheroid, but the oblateness will be less than 
when the fluid is homogeneous. 

669. The time of the earth's rotation, as well as the position of 
its axis, would change if any variation should take place in the 
distribution of the matter of the earth, or in case of the impact of 
a foreign body. 

If any portion of matter be, from any cause, made to approach 
the axis, its velocity will be diminished, and the velocity lost being 
imparted to the mass, will tend to accelerate the rotation. If any 
portion of matter be made to recede from the axis, the opposite 
effect will be produced, or the rotation will be retarded. In point 
of fact, the changes that take place in the position of the matter 
of the earth, whether from the washing of rains upon the sides of 
mountains, or evaporation, or any other known cause, are not suf- 
ficient ever to produce any sensible alteration in the circumstances 
of the earth's rotation on its axis. 



PHYSICAL THEORY OF PRECESSION AND NUTATION. 253 

670. It is ascertained from direct observation, that there has in 
reality been no perceptible change in the period of the earth's ro- 
tation since the time of Hipparchus, 120 years before the begin- 
ning of the present era. We may therefore conclude, a posteriori, 
that there has been no material change in the form and dimensions 
of the earth in this interval. 

671. Were the axis of the earth to experience any change of 
position with respect, to the matter of the earth, the latitudes of 
places would be altered. A motion of 200 feet might increase oi 
diminish the latitude of a place to the amount of 2", an angle which 
can be measured by modern instruments. Now ? in point of fact 7 
the latitudes of places have not sensibly varied since their first de- 
termination with accurate instruments ; therefore, in this interval 
the axis of the earth cannot have materially changed. Indeed y 
since the earth's surface and its internal strata are arranged sym- 
metrically with respect to the present axis of rotation, it is to be in- 
ferred that this axis is the same as that which obtained at the epoch 
when the matter of the earth changed from a fluid to a solid state 

672. The motions of the earth's axis, along with the whole body 
of the earth, which give rise to the Precession of the Equinoxes 
and Nutation, are consequences of the spheroidal form of the 
earth, inasmuch as they are produced by' the actions of the sun 
and moon upon that portion of the matter of the earth which lies 
on the outside of a sphere conceived to be described about the 
earth's axis. The physical theory of the phenomena in question 
is analogous to that of the retrogradation of the moon's nodes. The 
sun produces a retrograde movement of the points in which the 
circle described by each particle of the protuberant mass cuts the 
plane of the ecliptic, as it does of the moon's nodes ; ; the effect 
produced is, however, exceedingly small, by reason of the inertia 
of the interior spherical mass connected with the external mass 
upon which the action takes place. The moon, in like manner, 
occasions a retrograde movement of the nodes of the same parti- 
cles on the plane of its orbit. The actions of the sun and moon 
will not be the same each revolution of a particle. That of the 
sun will vary during the year with the angular distance of the sun 
from the node, (649 ;) and that of the moon will vary during each 
month with the distance of the moon from the node, and also 
during a revolution of the nodes of the moon's orbit by reason of 
the change in the inclination of the orbit to the equator. The 
mean effect of both bodies is the precession; the inequality re- 
sulting from the change in the sun's action during the year is the 
solar nutation ; and the inequality consequent upon the retrogra- 
dation of the moon's nodes is the lunar nutation, or the chief 
part of it : the change in the position of the equinox occasioned by 
the moon's revolution, never exceeds £ of a second of an arc; and the 
change of the obliquity of the ecliptic from this cause is still less. 



254 OF THE TIDES. 



CHAPTER XXIV. 

OF THE TIDES. 

673. The alternate rise and fall of the surface of the ocean 
twice in the course of a lunar day, or about 25 hours, is the phe- 
nomenon known bv the name of the Tides. The rise of the water 
is called the Flood Tide, and the fall the Ebb Tide. 

674. The interval between one high water and the next is. at a 
mean, half a mean lunar day, or 12h. 25m. 14s. Low water has 
place nearly, but not exactly, at the middle of this interval ; the 
tide, in general, employing nine or ten minutes more in ebbing than 
in flowing. As the interval between one period of high water and 
the second following one is a lunar day, or Id. Oh. 50m. 2Ss., the 
retardation in the time of high water from one day to another is 
50m. 28s., in its mean state. 

675. The time of high water is mainly dependent upon the po- 
sition of the moon, being always, at any given place, about the 
same length of time after the moon's passage over the superior or 
inferior meridian. As to the length of the interval between the 
two periods, at different places, in the open sea it is only from two 
to three hours ; but on the shores of continents, and in rivers, 
where the water meets with obstructions, it is very different at 
different places, and in some instances is of such length that the 
time of high water seems to precede the moon's passage. 

676. The height of the tide at high water is not always the 
same, but varies from day to day ; and these variations have an 
evident relation to the phases of the moon. It is greatest at the 
syzigies ; after which it diminishes and becomes the least at the 
quadratures.* 

677. The tides which occur near the syzigies, are called the 
Spring Tides ; and those which occur near the quadratures are 
called the Neap Tides. 

The highest of the spring tides is not that which has place 
nearest to new or full moon, but is in general the third following 
tide. In like manner the lowest of the neap tides is the tlnrd or 
fourth tide after the quadrature. 

The spring tides are, in general, about twice the height of the 
neap tides. At Brest, in France, the former rises to the height of 
19.3 feet, and the latter only to 9.2 feet. In the Pacific Ocean the 
highest of the tides of the syzigies is 5 feet, and the lowest of the 
tides of the quadratures is between 2 and 2.5 feet. 

678. The tides are also affected by the declinations of the sun 
and moon : thus, the highest spring tides in the course of the year 

* Bailj's Astronomical Tables and Formulae, p. 25. 



PHENOMENA OF THE TIDES. 255 

are those which occur near the equinoxes. The extraordinarily high 
tides which frequently occur at the equinoxes are, however, in 
part attributable to the equinoctial gales. Also, when the moon or 
the sun is out of the equator, the evening and morning tides differ 
somewhat in height. At Brest, in the syzigies of the summer sol- 
stice, the tides of the morning of the first and second day after the 
syzigy are smaller than those of the evening by 6.6 inches. They 
are greater by the same quantity in the syzigies of the winter sol- 
stice.* 

679. The distance of the moon from the earth has also a sensi- 
ble influence upon the tides. In general, they increase and dimin- 
ish as the distance increases and diminishes, but in a more rapid 
ratio. 

680. The daily retardation of the time of high water varies with 
the phases of the moon. It is at its minimum towards the syzigies, 
when the tides are at their maximum ; and it is then about 40m. 
But, towards the quadratures, when the tides are at their minimum, 
the retardation is the greatest possible ; and amounts to about lh. 
15m. 

The variation in the distance of the sun and moon from the earth, 
(and particularly the moon,) has an influence also on this retarda- 
tion. 

The daily retardation of the tides varies likewise with the decli- 
nation of the sun and moon.f 

681. The facts which have been detailed indicate that the tides 
are produced by the actions of the sun and moon upon the waters 
of the ocean ; but in a greater degree by the action of the moon. 
To explain them, let us suppose at first that the whole surface of 
the earth is covered with water. We remark, in the first place, 
that it is not the whole attractive force of the moon or sun which 
is effective in raising the waters of the ocean, but the difference in 
the actions of each body upon the different parts of the earth ; or, 
more precisely, that the phenomenon of the tides is a consequence 
of the inequality and non-parallelism of the attractive forces exert- 
ed by the moon, as well as by the sun, upon the different particles 
of the earth's mass. From this cause there results a diminution 
in the gravity of the particles of water at the surface, for a certain 
distance about the point immediately under the moon, and the point 
diametrically opposite to this, and an augmentation for a certain 
distance on the one side and the other of the circle 90° distant from 
these points, or of which they are the geometrical poles : in con- 
sequence of which the water falls about this circle and rises about 
these points. That the actions of the moon upon the different 
parts of the earth's mass are really unequal is evident, from the 
fact, that these parts are at different distances from the moon. To 



Laplace's System of the World. t Baily's Tables and Formul®, p. 26. 



256 



OF THE TIDES. 



show that the inequality will give rise to the results just noted, let 
us suppose that the circle acbd (Fig. 122) represents the earth, and 
M the place of the moon ; then a will be the point of the earth's 
Fig. 122. surface directly under the moon, b the 

I point diametrically opposite to this, and 

the right line dc perpendicular to MO 
will represent the circle traced on the 
earth's surface 90° distant from a and b. 
Now, the attraction of the moon for the 
general mass of the earth is the same as 
if the whole mass were concentrated at 
the centre 0. But the centre of the 
earth is more distant from the moon 
than the point 'a at the surface. It fol- 
lows, therefore, that a particle of matter 
situated at a will be drawn towards the 
moon with a proportionally greater force 
than the centre, or than the general mass 
of the earth. Its gravity or tendency 
towards the earth's centre will therefore 
be diminished by the amount of this ex- 
cess. On the other hand, the centre is 
nearer to the moon than the point 6. It 
is therefore attracted more strongly than 
a particle at b. The excess will be a 



I 






>\ 


7 *K 




y Am 




a 


7 


*7 

/ 




\ i 




h 1 



M 



be the same 
centre by the 



as 
same 



if the 
force 



force tending to draw the centre away 
from the particle ; and the effect will 
particle were drawn away from the 
acting in the opposite direction. The 
result then is, that this particle has its gravity towards the earth's 
centre diminished, as well as the particle at a. If now we consider 
a particle at some point t near to a, the moon's action upon 
it (tr) may be considered as taking effect partially in the direction 
tk parallel to OM, and partially in the direction of the tangent or 
horizontal line ts. The component (ts) in the latter direction, will 
have no tendency to alter the gravity of the particle towards the 
earth's centre. The component (sr) in the direction tk, will obvi- 
ously be less than the actual force of attraction tr ; and the dif- 
ference will be greater in proportion as the particle is more remote 
from a. But this component will decrease gradually from a, while 
the attraction for the centre is less than for a by a certain finite differ- 
ence : it is plain, therefore, that the component in question will be 
greater than the attraction for the centre, in the vicinity of the point 
a, and for a certain distance from it in all directions. The gravity 
of the particles will therefore be diminished for a certain distance 
from this point. In a similar manner it may be shown that it will 
also be diminished for a certain distance from the point b. Let us 
now consider a particle at c, 90° from the points a and b. The at- 



PHYSICAL THEORY OF THE TIDES. 257 

traction of the moon for it will take effect in the two directions cl 
and cO. The force in the latter direction alone will alter the grav- 
ity of the particle ; and this, it is plain, will increase it. The same 
effect will extend to a certain distance from c in both directions. 

A strict mathematical investigation would show that the gravity- 
is diminished for a distance of 55° from a and b in all directions ; 
and is augmented for a distance of 35° on each side of the circle 
dc, 90° distant from the points a and b. These distances are rep- 
resented in the Figure. 

This may be easily made out by means of the expression for the radial disturb- 

Ttl 

ing force of the sun in its action upon the moon, (643,) viz. — y (1 — 3cos 2 ^). If 

we consider m as denoting the mass of the moon, a the moon's distance from the 
earth's centre, y the distance of a particle of matter at some point t of the earth's 
surface from the earth's centre, and <£ the angular distance or elongation (MO*) 
of the same particle from the moon, as seen from the centre of the earth, it will ex- 
press the change in the gravity of a particle at the earth's surface, produced by the 
moon's action. The points a and b will answer to conjunction and opposition, and 
the points c and d to the quadratures. Now we have already seen (643) that the 
gravity of the moon is increased at the quadratures, and for 35° on each side of 
them ; and diminished at the syzigies, and 55° from them in both directions. It fol- 
lows, therefore, that the same is true for particles of matter at the earth's surface. 

In consequence of the earth's diurnal rotation, the parts of the 
surface, at which the rise and fall of the water will take place, will 
be continually changing. Were the entire rise and fall produced 
instantaneously, the points of highest water would constantly be the 
precise points in which the line of the centres of the moon and 
earth intersects the surface, and it would always be high water on 
the meridian passing through these points, both in the hemisphere 
where the moon is, and in the opposite one. On the west side of 
this meridian, the tide would be flowing ; on the east side of it, it 
would be ebbing ; and on the meridian at right angles to the same, 
it would be low water. But it is plain that the effects of the moon's 
action will not be instantaneously produced, and therefore that the 
points of highest water will fall behind the moon. It appears from 
observation, that in the open sea the meridian of high water is about 
30° to the east of the moon. 

The great tide wave thus raised by the moon, and which follows 
it in its diurnal motion, will be a mere undulation, or alternate rise 
and fall of the water, without any progressive motion, if, as we have 
supposed, it is nowhere obstructed by shallows, islands, or the 
shores of continents. 

682. It is evident that the sun will produce precisely similar 
effects with the moon, and will raise a tide wave similar to the 
lunar tide wave, which will follow it in its diurnal motion. 

683. To 6how that the effects of the sun are less in degree than those of the 
moon, let us take the general expression for the change of the moon's gravity* 
arising from the action of the sun, namely, 

^Xy(l-3cos2*) . . . (a)| 
33 



258 OF THE TIDES 

in which m denotes the mass of the sun, a its distance, (the mean distance of the 
moon being 1 taken as 1,) y the distance of the moon in its given position, and <p its 
elongation from the sun, as seen from the earth's centre. This formula will serve 
to express the change in the gravity of a particle of matter upon the earth's sur- 
face, produced by the sun's action, if we take m = the mass of the sun, as before, 
a = its distance expressed in terms of the radius of the earth as unity, y = the 
distance of the particle from the centre of the earth, and <p = its elongation from 
the sun, as seen from the earth's centre. If we designate the corresponding quan- 
tities for the moon by m! , a', y, <f>, we shall have for the change of the gravity of 
a particle, produced by the moon's action, 

!^Xy(l — 3cos20) . . . (6). 

For particles at equal elongations from the sun and moon, we shall have the 
same in expressions (a) and (b), and y may be regarded as the same without ma- 
terial error. For such particles, then, the alterations of the gravity, produced 
by the sun and moon, will bear the same ratio to each other as the quantities 

— and — . Now, if we give to m, m', a, a', their values, we shall find that the 

latter quantity is nearly three times greater than the former. Accordingly, the 
effect of the moon's action, at corresponding elongations of the particles, and there- 
fore generally, is nearly three times greater than that of the sun. 

684. The actual tide will be produced by the joint action of the 
sun and moon, or it may be regarded as the result of the combina- 
tion of the lunar and solar tide waves. 

At the time of the syzigies, the action of the sun and moon will 
be combined in producing the tides, both bodies tending to produce 
high as well as low water at the same places. But at the quadra- 
tures they will be in opposition to each other, the one tending to 
raise the surface of the water where the other tends to depress it, 
and vice versa. The tides should, therefore, be much higher at 
the syzigies than at the quadratures. 

Between the syzigies and the quadratures the two bodies will 
neither directly conspire with each other, nor directly oppose each 
other, and tides of intermediate height will have place. The points 
of highest water will also, in the configuration supposed, neither 
be the vertices of the lunar nor of the solar tide wave, but certain 
points between them. This circumstance will occasion a variation 
in the length of the interval between the time of the moon's pas- 
sage and the time of high water. 

685. The effect of the moon's action being to that of the sun's 
nearly as 3 to 1, (683,) the spring tides will be to the neap tides 
nearly as 2 to 1 . For, let x — the effect of the moon, and y = 
the effect of the sun : then the ratio of x + y to x — y will be the 
ratio of the heights of the spring and neap tides. Now, 

x = 3y, and thus X -±l = ^±l = 2 , 
x-y Sy-y 

This result is conformable to observation. 

686. The height of the tide, as well as the interval between the 
time of high water and that of the moon's meridian passage, will 
vary not only with the elongation of the moon from the sun, but 



MODIFICATIONS OF THE GENERAL PHYSICAL THEORY. 259 

•also with the distance and declination of the moon and sun. For, 
expressions {a) and (b) show that the intensities of the moon's and 
sun's actions vary inversely as the cube of their distance ; and the 
changes of the declinations of the two bodies must be attended 
with a change both in the absolute and relative situation of the 
vertices of the lunar and solar tide waves. 

687. The laws of the tides, which would obtain on the hypothe- 
sis of the earth being covered entirely with water, are found to 
correspond only partially with those of the actual tides. The 
continents have a material influence upon the formation and pro- 
pagation of the tide wave. 

688. Professor Whewell infers, from a careful discussion of a 
great number of observations upon the tides, that the tide of the 
Atlantic Ocean is, for the most part, produced by a derivative tide 
wave, sent off from the great wave which in the Southern Ocean 
follows the moon in its diurnal motion around the earth. This 
wave advances more rapidly in the open sea than along the coasts, 
where it meets with obstructions. 

Where portions of the tide wave, extending from one point of 
the coast to another, become detached, and advance into a narrow 
space, particularly high tides will occur. In this way (as it is sup 
posed) it happens that the tide rises at certain places in the Bay 
of Fundy, to the height of 60 or 70 feet 

689. In channels peculiar tides occur in consequence of the 
meeting of the waves which enter the channels at their two ex- 
tremities. Where the two waves meet in the same state, unusually 
high tides occur. This is observed to be the case at some points 
in the Irish Channel. In the port of Batsha, in Tonquin, the tides 
arrive by two channels, of such lengths that the two waves meet 
in opposite states, or that the flood tide arrives by one channel just 
as the ebb tide begins to leave by the other, and the consequence 
is that there is neither high nor low water. 

This is the case when the moon is in the equator. When she 
has a northern or southern declination, there is a small rise and 
fall of the water once in a lunar day, owing to the inequality of the 
morning and evening tides of the open sea. 

690. Lakes and inland seas have no perceptible tides, for the 
reason that their extent is not sufficient to admit of any sensible 
inequality of gravity, as the result of the action of the moon. 

691. The tides experienced in rivers and seas communicating 
with the ocean, are not produced by the direct actions of the sun 
and moon, but are waves propagated from the great wave of the 
open sea. 

In rivers of considerable length, the ascending tides are encoun- 
tered by those which are returning, so that a great variety of tides 
occur along their shores, 

692. The mean interval between noon and the time of high 
water at any port, on the day of new or full moon, is called the 



860 OF THE TIDES, 

Establishment of that port. It will be, approximately, the inter- 
val between the time of the meridian passage of the moon and the 
time of high water on any day of the month. To obtain this in- 
terval for a given day more nearly, it is necessary to correct the 
establishment for the effects of the change of the distance and de- 
clination of the snn and moon, and of the change in the elongation 
of the moon from the sun. When it has been determined, by add- 
ing it to the time of the meridian passage of the moon ? we have the 
time of the next high water. 



PART IV. 

ASTRONOMICAL PROBLEMS. 



EXPLANATIONS OF THE TABLES. 

The Tables which form a part of this work, and which are em- 
ployed in the resolution of the following Problems, consist of Ta- 
bles of the Sun, Tables of the Moon, Tables of the Mean Places 
of some of the Fixed Stars, Tables of Corrections for Refraction, 
Aberration, and Nutation, and Auxiliary Tables. 

The Tables of the Sun, which are from XVII to XXXIV, in- 
elusive, are, for the most part, abridged from Delambre's Solar Ta- 
bles. The mean longitudes of the sun and of his perigee for the 
beginning of each year, found in Table XVIII, have been com- 
puted from the formulas of Prof- Bessel, given in the Nautical Al- 
manac of 1837. The Table of the Equation of Time was reduced 
from the table in the Connaissance des Terns of 1810, which is 
more accurate than Delambre's Table, this being in some instances 
liable to an error of 2 seconds. The Table of Nutation (Table 
XXVII) was extracted from Francoeur's Practical Astronomy. 
The maximum of nutation of obliquity is taken at 9".25. The 
Tables of the Sun will give the sun's longitude within a frac- 
tion of a second of the result obtained immediately from De- 
lambre^s Tables, as corrected by BesseL The Tables of the 
Moon, which are from XXXIV to LXXXV, inclusive, are 
abridged and computed from Burckhardt's Tables of the Moon. 
To facilitate the determination of the hourly motions in longi- 
tude and latitude, the equations of the hourly motions have all 
been rendered positive, like those of the longitude. Some few new 
tables have been computed for the same purpose. The longitude 
and hourly motion in longitude will very rarely differ from the re- 
sults of Burckhardf s Tables more than 0".5, and never as much 
as V\ The error of the latitude and hourly motion in latitude will 
be still less. The other tables have been taken from some of the 
most approved modern Astronomical Works. (For the principles 
of the construction of the Tables, see Chap, IX.) 

Before entering upon the explanation of each of the tables, it 
will be proper to define a few terms that will be made use of in the 
sequel. 

The given quantity with which a quantity is taken from a table, 
is called the Argument of this quantity. 



262 ASTROJSOMICAL PROBLEMS. 

The angular arguments are expressed in some of the tables ac- 
cording to the sexagesimal division of the circle. In others, they 
are given in parts of the circle supposed to be divided into 10O ? 
1000, or 10000, &c, pan's. 

Tables are of Single or Double Entry, according as they con- 
tain one or two arguments. The Epoch of a table is the instant 
of time for which the quantities given by the table are computed. 
By the Epoch of a quantity, is meant the value of the quantity 
found for some chosen epoch, from which its value at other epochs 
is to be computed by means of its known rate of variation. 

Table I, contains the latitudes and longitudes from the meridian 
of Greenwich, of various conspicuous places in different parts of 
the earth. The longitudes serve to make known the time at any 
one of the places in the table, when that at any of the others is 
given. The latitude of a place is an important element in various 
astronomical calculations. 

Table II, is a table of the Elements of the Orbits of the Planets, 
with their secular variations, which serve to make known the ele- 
ments at any given epoch different from that of the table. From 
these the elliptic places of the planets at the given epoch may be 
computed. 

Table III, is a similar table for the Moon. 

Tables IV, V, VI, VII, require no explanation. 

Table VIII, gives the mean Astronomical Refractions ; that is r 
the refractions which have place when the barometer stands at 30 
inches, and the thermometer of Fahrenheit at 50°. 

Table IX, contains the corrections of the Mean Refractions for 
-f 1 inch in the barometer, and — 1° in the thermometer, from 
which the corrections to be applied, at any observed height of the 
barometer and thermometer, are easily derived. 

Table X, gives the Parallax of the Sun for any given altitude on 
a given day of the year ; for reducing a solar observation made at 
the surface of the earth to what it would have been, if made at the 
centre. 

Table XI, is designed to make known the Sun's Semi-diurnal 
Arc, answering to any given latitude and to any given declination 
of the sun ; and thus the time of the sun's rising and setting, and 
the length of the day. 

Table XII, serves to make known the value of the Equation of 
Time, with its essential sign, which is to be applied to the apparent 
time to convert it into the mean. If the sign of the equation taken 
from the table be changed, it will serve for the conversion of mean 
time into apparent. This table is constructed for the year 1840. 

Table XIII, is to be used in connection with Table XII, when 
the given date is in any other year than 1840. It furnishes the 
Secular Variation of the Equation of Time, from which the pro- 
portional part of its variation in the interval between the given date 
and the epoch of Table XII is easily derived. 



EXPLANATION OF THE TABLES. 263 

Table XIV, contains certain other Corrections to be applied to 
the equation of time taken from Table XII, when its exact value, 
to within a small fraction of a second, is desired. 

Table XV, gives the Fraction of the Year corresponding to each 
date. This table is useful when quantities vary by known and uni- 
form degrees, in deducing their values at any assumed time from 
their values at any other time. 

Table XVI, is for converting Hours, Minutes, and Seconds into 
decimal parts of a Day. 

Table XVII, is for converting Minutes and Seconds of a degree 
into the decimal division of the same. It will also serve for the 
conversion of minutes and seconds of time into decimal parts of an 
hour. 

The last two tables will be found frequently useful in arithmeti- 
cal operations 

Table XVIII, is a table of Epochs of the Sun's Mean Longi- 
tude, of the Longitude of the Perigee, and of the Arguments for 
finding the small equations of the Sun's place. They are all cal- 
culated for the first of January of each year, at mean noon on the 
meridian of Greenwich. Argument I. is the mean longitude of the 
Moon minus that of the Sun ; Argument II. is the heliocentric 
longitude of the Earth ; Argument III. is the heliocentric longi- 
tude of Venus ; Argument IV. is the heliocentric longitude of 
Mars ; Argument V. is the heliocentric longitude of Jupiter ; Ar- 
gument VI. is the mean anomaly of the Moon ; Argument VII. is 
the heliocentric longitude of Saturn ; and Argument N is the sup- 
plement of the longitude of the Moon's Ascending Node. Argu- 
ment I. is for the first part of the equation depending on the action 
of the Moon. Arguments I. and VI. are the arguments for the re- 
maining part of the lunar equation. Arguments II. and III. are for 
the equation depending on the action of Venus ; Arguments II. 
and IV. for the equation depending on the action of Mars ; Argu- 
ments II. and V. for the equation depending on the action of Ju- 
piter ; and Arguments II. and VII. for the equation depending on 
the action of Saturn. Argument N is the argument for the Nuta- 
tion in longitude : it is also the argument for the Nutation in right 
ascension, and of the obliquity of the ecliptic. 

Table XIX, shows the Motions of the Sun and Perigee, and the 
variations of the arguments, in the interval between the beginning 
of the year and the first of each month. 

Table XX, shows the Motions of the Sun and Perigee, and the 
variations of the arguments from the beginning of any month to the 
beginning of any day of the month ; also the same for Hours. 

Table XXI, gives the Sun's Motions for Minutes and Seconds. 
Tables XVIII to XXI, inclusive, make known the mean longitude 
of the Sun from the mean equinox, at any moment of time. 

Table XXII, Mean Obliquity of the Ecliptic for the beginning 



264 ASTRONOMICAL PROBLEMS. 

of each year contained in the table. It is found for any interme- 
diate time by simple proportion. 

Tables XXIII, and XXIV, furnish the Sun's Hourly Motion 
and Semi-diameter. 

Table XXV, is designed to make known the Equation of the 
Sun's Centre. When the equation has the negative sign, its sup- 
plement to 12s. is given : this is to be added along with the other 
equations of longitude, and 12s. are to be subtracted from the sum. 

The numbers in the table are the values of the equation of the 
centre, or of its supplement, diminished by 46". 1. This constant 
is subtracted from each value, to balance the different quantities 
added to the other equations of the longitude, in order to render 
them affirmative. The epoch of this table is the year 1840. 

Table XXVI, gives the Secular Variation of the Equation of the 
Sun's Centre, from which the proportional part of the variation in 
the interval between the given date and the year 1840, may be 
derived. 

Table XXVII, is for the Nutation in Longitude, Nutation in 
Right Ascension, and Nutation of the Obliquity of the Ecliptic. 
The nutation in longitude and nutation in right ascension, serve to 
transfer the origin of the longitude and right ascension from the 
mean to the true equinox. And the nutation of obliquity serves to 
change the mean into the true obliquity. 

Tables XXVIII to XXXIII, inclusive, give the Equations of 
the Sun's Longitude, due respectively to the attractions of the 
Moon, Venus, Jupiter, Mars, and Saturn. 

Table XXXIV, is for the variable part of the Sun's Aberration, 
The numbers have all been rendered positive by the addition of 
the constant 0".3. 

Table XXXV, contains the Epochs of the Moon's Mean Longi- 
tude, and of the Arguments of the equations used in determining 
the True Longitude and Latitude of the Moon. They are all cal- 
culated for the first of January of each year, at mean noon on the 
meridian of Greenwich. The Argument for the Evection is di- 
minished by 30' ; the Anomaly by 2° ; the Argument for the Va- 
riation by 9°, and the mean longitude by 9° 45 ; ; and the Supple- 
ment of the Node is increased by T . This is done to balance the 
quantities which are added to the different equations in order to 
render them affirmative. 

Tables XXXVI to XL, inclusive, give the Motions of the Moon, 
and the variations of the arguments, for Months, Days, Hours, 
Minutes, and Seconds ; and, together with Table XXXV, are for 
finding the Moon's Mean Longitude and the 'Arguments, at any 
assumed moment of time. 

Tables XLI to LIII, inclusive, give the various Equations of 
the Moon's Longitude. It is to be observed with respect to Table 
XLI, that the right hand figure of the argument is supposed to be 
dropped. But when the greatest attainable accuracy is desired, it 



EXPLANATION OF THE TABLES. 265 

can be retained, and a cipher conceived to be written after the 
numbers in the columns of Arguments in the table. In Tables 
L, LI, LII, and LV, the degrees will be found by referring to the 
head or fooj of the column. (See Problem II., note 2.) 

Table LIV is for the Nutation of the Moon's Longitude. 

Tables LV to LIX, inclusive, are for finding the Latitude of 
the Moon. 

Tables LX to LXIII, inclusive, are for the Equatorial Paral- 
lax of the Moon. 

Table LXIV furnishes the Reductions of Parallax and of the 
Latitude of a Place. The reduction of parallax is for obtaining 
the parallax at any given place from the equatorial parallax. The 
reduction of latitude is foi reducing the true latitude of a place, as 
determined by observation, to the corresponding latitude on the 
supposition of the earth being a sphere. The ellipticity to which 
the numbers in the table correspond is -3 ^- ¥ . 

Tables LXV and LXVI, Moon's Semi-diameter, and the Aug- 
mentation of the Semi-diameter depending on the altitude. 

Tables LXVII to LXXXV, inclusive, are for finding the 
Hourly Motions of the Moon in Longitude and Latitude. 

Table LXXXVI, Mean New Moons, and the Arguments for the 
Equations for New and Full Moon, in January. The time of 
mean new moon in January of each year has been diminished by 
15 hours, the sum of the quantities which have been added to the 
equations in Table LXXXIX. Thus, 4h. 20m. has been added 
to equation I. ; lOh. 10m. to equation II. ; 10m. to equation III.; 
and 20m. to equation IV. 

Tables LXXXVII and LXXXVIII, are used with the preced- 
ing in finding the Approximate Time of Mean New or Full Moon 
in any given month of the year. 

Table LXXXIX furnishes the Equations for finding the Ap- 
proximate Time of New or Full Moon. 

Table XC contains the Mean Right Ascensions and Declina- 
tions of 50 principal Fixed Stars, for the beginning of the year 
1840, with their Annual Variations. 

Table XCI is for finding the Aberration and Nutation of the 
Stars in the preceding catalogue. 

Table XCII contains the Mean Longitudes and Latitudes of 
some of the principal Fixed Stars, for the beginning of the year 
1840, with their Annual Variations. 

Tables XCIII, XCIV, XCV, Second, Third, and Fourth 
Differences. These tables are given to facilitate the determina- 
tion, from the Nautical Almanac, of the moon's longitude or lati- 
tude for any time between noon and midnight. 

Table XCVI, Logistical Logarithms. This table is convenient 
in working proportions, when the terms are minutes and seconds, 
or degrees and minutes, or hours and minutes, — especially when 
the first term is lh. or 60m. 

34 



266 ASTRONOMICAL PROBLEMS. 

To find the logistical logarithm of a number composed of min- 
utes and seconds, or degrees and minutes, of an arc ; or of min- 
utes and seconds, or hours and minutes, of time. 

1 . If the number consists of minutes and seconds, afc the top of 
the table seek for the minutes, and in the same column opposite 
the seconds in the left-hand column will be found the logistical 
logarithm. 

2. If the number is composed of hours and minutes, the hours 
must be used as if they were minutes, and the minutes as if they 
were seconds. 

3. If the number is composed of degrees and minutes, the de- 
grees must be used as if they were minutes, and the minutes as if 
they were seconds. 

To find the logistical logarithm of a number less than 3600. 

Seek in the second line of the table from the top the number 
next less than the given number, and the remainder, or the com- 
plement to the given number, in the first column on the left : then 
in the column of the first number, and opposite the complement, 
will be found the logistical logarithm of the sum. Thus, to ob- 
tain the logarithm of 1531, we seek for the column of 1500, and 
opposite 31 we find 3713. 



PROBLEM I. 



To work, by logistical logarithms, a proportion the terms of which 
are degrees and minutes, or minutes and seconds, of an arc ; or 
hours and minutes, or minutes and seconds, of time. 

With the degrees or minutes at the top, and minutes or seconds 
at the side, or if a term consists of hours and minutes, or minutes 
and seconds, with the hours or minutes at the top, and minutes 
or seconds at the side, take from Table XCVI. the logistical loga- 
rithms of the three given terms ; add together the logistical loga- 
rithms of the second and third terms and the arithmetical comple- 
ment of that of the first term, rejecting 10 from the index.* The 
result will be the logistical logarithm of the fourth term, with 
which take it from the table. 

Note 1 . The logistical logarithm of 60' is 0. 

Note 2. If the second or third term contains tenths of seconds, 
(or tenths of minutes, when it consists of degrees and minutes,) 
and is less than 6', or 6°, multiply it by 10, and employ the loga- 
rithm of the product in place of that of the term itself. The 

* Instead of adding the arithmetical complement of the ogarithm of the first 
term, the logarithm itself may be subtracted from the sum of the logarithms of the 
other two terms. 



TO TAKE OUT A QUANTITY FROM A TABLE. 26"? 

result obtained by the table, divided by 10, will be the fourth term 
of the proportion, and will be exact to tenths. 

Note 3. If none of the terms contain tenths of minutes or sec- 
onds, and it is desired to obtain a result exact to tenths, diminish 
the index, of the logistical logarithm of the fourth term by 1, and 
cut off the right-hand figure of the number found from the table, 
for tenths. 

Exam. 1. When the moon's hourly motion is 30' 12", what is 
its motion in 16m. 24s. ? 

As 60m 

: 30' 12" .... 2981 
: : 16m. 24s. . . . 5633 



: 8' 15" .... 8614 
2. If the moon's declination change 1° 31' in 12 hours, what 
will be the change in 7h. 42m. ? 

As 12h. . . . ar. co. 9.3010 

: 1° 31' . . . 1.5973 

:: 7h. 42m. , . . 8917 



: 0° 58' . . . 1.7900 

3. When the moon's hourly motion in latitude is 2' 26".8, what 
is its motion in 36m. 22s. ? 
2' 26".8 
60 



146".8 
10 



As 60m. . . 

1468 . . : 1468" . . 3896 
: : 36m. 22s. . 2174 



: 890" . . 6070 

Ans. 1' 29".0. 

4. When the sun's hourly motion in longitude is 2' 28", what 
is its motion in 49m. lis. ? Ans. 2' 1". 

5. If the sun's declination change 16' 33" in 24 hours, what 
will be the change in 14h. 18m. ? Ans. 9' 52". 

6. If the moon's declination change 54".7inone hour, what will 
be the change in 52m. 18s. ? Ans. 47".7. 



PROBLEM II. 

To take from a table the quantity corresponding to a given value 
of the argument, or to given values of the arguments of the 
table. 



268 ASTRONOMICAL PROBLEMS. 

Case 1. When quantities are given in the table for each sign 
and degree of the argument. 

With the signs of the given argument at the top or bottom, and 
the degrees at the side, (at the left side, if the signs are found at 
the top; at the right side, if they are found at the bottom,) take out 
the corresponding quantity. Also take the difference between this 
quantity and the next following one in the table, and say, 60' : this 
difference : : odd minutes and seconds of given argument : a fourth 
term. This fourth term, added to the quantity taken out, when the 
quantities in the table are increasing, but subtracted when they are 
decreasing, will give the required quantity. 

Note 1. When the quantities change but little from degree to 
degree of the argument, the required quantity may often be esti- 
mated, without the trouble of stating a proportion. 

Note 2. In some of the tables the degrees or signs of the quan- 
tity sought, are to be had by referring to the head or foot of the col- 
umn in which the minutes and seconds are found. (See Tables 
L, LI, LII, and LV.) The degrees there found are to be taken, 
if no horizontal mark intervenes ; otherwise, they are to be in- 
creased or diminished by 1°, or 2°, according as one or two marks 
intervene. They are to be increased, or diminished, according as 
their number is less or greater than the number of degrees at the 
other end of the column. 

Note 3. If, as is the case with some of the tables, the quantities 
in the table have an algebraic sign prefixed to them, neglect the 
consideration of the sign in determining the correction to be applied 
to the quantity first taken out, and proceed according to the rule 
above given. The result will have the sign of the quantity first 
taken out. It is to be observed, however, that if the two consecu- 
tive quantities chance to have opposite signs, their numerical sum 
is to be taken instead of their difference ; also that the quantity 
sought will, in every such instance, be the numerical difference 
between the correction and the quantity first taken out, and, ac- 
cording as the correction is less or greater than this quantity, is to 
be affected with the same or the opposite sign. 

Exam. 1. Given the argument 7 s - 6° 24' 36", to find the corre- 
sponding quantity in Table L. 

7 s - 6° gives 0° 43' 17".4. 
The difference between 0° 43' 17". 4 and the next following quan- 
tity in the table is 1' 7". 3. 

60' : 1' 7".3 : : 24' 36" : 27".6.* 

* The student can work the proportion, either by the common method, or by lo- 
gistical logarithms, as he may prefer. In working this and all similar proportions 
by the arithmetical method, the seconds of the argument may be converted into 
the equivalent decimal part of a minute by means of Table XVII, (using the sec 
onds as if they were minutes.) It will be sufficient to take the fraction to the 
nearest tenth. 



TO TAKE OUT A QUANTITY FROM A TABLE. 269 

From 0° 43' 17" A 
Take 27 .6 



42 49 .8 
2. Given the argument 2 s - 18° 41' 20", to find the corresponding 
quantity in Table XXV. 

2 s - 18° gives 1° 52' 32".5. 
The difference between 1° 52' 32". 5 and the next following 
quantity in the table is 21 ".8. 

60' : 21".8 : : 41' 20" : 15".0. 

To 1° 52' 32".5 
Add 15 .0 



1 52 47 .5 
3. Given the argument 9 8 ' 2° 13' 33", to find the correspond- 
ing quantity in Table XII. 

9 s - 2° gives 29.8s. 
The arithmetical sum of 29.8s. and the next following quantity 
in the table is 30.4s. 

60' : 30.4s. : : 13° 33' : 6.9s. 

From 29.8s. 
Take 6.9 



22.9s. 
Ans. — 22.9s. 

4. Given the argument 5 s - 8° 14' 52", to find the corresponding 
quantity in Table LII. Ans. 12' 36".0. 

5. Given the argument 11 s - 11° 23' 10", to find the correspond- 
ing quantity in Table LVI. Ans. 11' 48".0. 

6. Given the argument s - 26° 20', to find the corresponding 
quantity in Table XII. Ans. - 41 8 ,0. 

Case 2. When the argument changes in the table by more or 
less than 1° ; or when it is given in lower denominations than 
signs. 

Take out of the table the quantity answering to the number in 
the column of arguments next less than the given argument. Take 
the difference between this quantity and the next following one, 
and also the difference of the consecutive values of the argument 
inserted in the table, and say, difference of arguments : difference 
of quantities : : excess of the given argument over the value next 
less in the table : a fourth term. This fourth term applied to the 
quantity first taken out, according to the rule given in the prece- 
ding case, will give the quantity sought. 

Note. In some of the tables the columns entitled Diff. are made 
up of the differences answering to a difference of 10' in the argu- 
ment In obtaining quantities from these tables, it will be found 
more convenient to take for the first and second terms of the pro- 



270 ASTRONOMICAL PROBLEMS. 

portion, respectively, 10', and the difference furnished by the table, 
and work the proportion by the arithmetical method. (See note at 
bottom of page 268.) 

Exam. 1. Given the argument s - 24° 42' 15", to find the cor- 
responding quantity in Table LI. 

s - 24° 30' gives 9° 47' 14".3. 
The difference between 9° 47' 14". 3 and the next following 
quantity = 3 x 63".0 = 189" .0. The argument changes by 30'. 
And the excess of s - 24° 42' 15" over s - 24° 30', is 12 15". Thus, 

30' : 189".0 : : 12' 15" : 77".2. 
But the correction may be found more readily by the following 
proportion : 

10' : 63".0 : : 12'.25 : 77".2. 

To 9° 47' 14".3 
Add 77 .2 



9 48 31 .5 
2. Given the argument 1° 12', to find the corresponding quan- 
tity in Table VIII. 

1° 10' gives 23' 13", 
and 5' : 33" : : 2' : 13" the correction. 

From 23' 13" 
Take 13 



23 

3. Given the argument 6 s - 6° 7' 23", to find the corresponding 
quantity in Table LV. Ans. 90° 20' 53".5. 

4. Given the argument 49° 27', to find the corresponding quan- 
tity in Table LXIV. Ans. 11' 19".8. 

Case 3. When the argument is given in the table in hundredth, 
thousandth, or ten thousandth parts of a circle. 

The required quantity can be found in this case by the same 
rule as in the preceding] but it can be had more expeditiously by 
observing the following rules. If the argument varies by 10, mul- 
tiply the difference of the quantities between which the required 
quantity lies by the excess of the given argument over the next less 
value in the table, and remove the decimal point one figure to the 
left ; the result will be the correction to be applied to the quantity 
taken out of the table. The same rule will apply in taking quan- 
tities from tables in which the differences answering to a change of 
10 in the argument are given, although the argument should actu- 
ally change by 50 or 100. If the argument changes by 100, mul- 
tiply as above, and remove the decimal point two figures to the left. 
When the common difference of the arguments is 5, proceed as if 
it were 10, and double the result. In like manner, when the com- 
mon difference is 50, proceed as if it were 100, and double the 
result. 



TO TAKE OUT A QUANTITY FROM A TABLE. 271 

Exam. 1. Given the argument 973, to find the corresponding 
quantity in Table XLV, column headed 13. 
970 gives 23".5. 
The difference is \" .2, and the excess 3. 

1".2 From 23".5 

3 Take .4 

' Corr. .36 23 .1 

2. Given the argument 4834, to find the corresponding quantity 
in Table XLII, column headed 5. 

4800 gives 2' 3".7. 
The difference is 6". 8, and the excess 34. 
6".8 
34 

From 2' 3".7 

2.312 . . . Take 2 .3 



2 1 .4 

3. Given the argument 5444, to find the corresponding quan- 
tity in Table XLI. Ans. 15' 37".7. 

4. Given the argument 4225, to find the corresponding quan- 
tity in Table XLIII, column headed 8. Ans. 0' 47". 2. 

Case 4. When the table is one of double entry, or quantities are 
taken from it by means of two arguments. 

Take out. of the table the quantity answering to the values of 
the arguments of the table next less than the given values ; and 
find the respective corrections to be applied to it, due to the ex- 
cess of the given value of each argument over the next less value 
in the table, by the general rule in the preceding case. These 
corrections are to be added to the quantity taken out, or subtracted 
from it, according as the quantities increase or decrease with the 
arguments. 

Note 1. If the tenths of seconds be omitted, the corrections 
above mentioned can be estimated without the trouble of stating a 
proportion, or performing multiplications. 

Note 2. The rule above given may, in some rare instances, give 
a result differing; a few tenths of a second from the truth. The 
following rule will furnish more exact results. Find the quanti- 
ties corresponding, respectively, to the value of the argument at 
the top next less than its given value and the other given argu- 
ment, and to the value next greater and the other given argument. 
Take the difference of the quantities found, and also the difference 
of the corresponding arguments at top, and say, difference of argu- 
ments : difference of quantities : : excess of given value of the 
argument at the top over its next less value in the table : a fourth 
term. This fourth term added to the quantity first found, if it is 
less than the other, but subtracted from it, if it is greater, will give 
the required quantity. The error of the first rule may be dimin- 



272 ASTRONOMICAL PROBLEMS. 

ished without any extra calculation, by attending to the difference 
of the quantities answering to the value of the argument at the 
side next greater than its given value and the values of the other 
argument between which its given value lies. 

Exam. 1 . Given the argument 64 at the top and 77 at the side, 
to find the corresponding quantity in Table LXXXI. 
50 and 70 give 47".7. 

The difference between 47".7 and the next quantity below it 
is 1".4. The excess of 77 over 70 is 7, and the argument at the 
side changes by 1 0. 



Corr. due excess 7 



1".4 






7 








From 


47".7 


.98, or 1".0. 


Take 


1 .0 



Quantity corresponding to 50 and 77, 46 .7 
The difference between 47". 7 and the adjacent quantity in the 
next column on the right is 3". 3. The excess of 64 over 50 is 14, 
and the argument at the top changes by 50. 
3".3 
14 

.462 
2 

From 46" .7 



Corr. due excess 14, .924 Take .9 



45 .8 
2. Given the argument 223 at the top and 448 at the side, to 
find the corresponding quantity in Table XXX. 
220 and 440 give 16".0. 
The difference between 16".0 and the quantity next below it 
is 2".2. 

2".2 
8 



2) 1.76 



From 16".0 
Corr. for excess 8, .88, or 0".9. Take .9 



Quantity corresponding to 220 and 448, 15 .1 
The difference between 16".0 and the adjacent quantity in the 
next column on the right is 0".7. 
0".7 
3 

To 15".l 

Coir, for excess 3, .21 Add .2 

15.3 



TO CONVERT DEGREES, MINUTES, ETC., INTO TIME. 273 

3. Given the argument 472 at the top and 786 at the side, to 
find the corresponding quantity in Table XXXI. 

Ans. 9".7. 

4. Given the argument 620 at the top and 367 at the side, to 
find the corresponding quantity in Table LXXXI. 

Ans. 55". 2. 

5. Given the argument 348 at the top and 932 at the side, to 
find (by the rule given in Note 2) the corresponding quantity in 
Table XXXII. Ans. 15".4. 



PROBLEM III. 

To convert Degrees, Minutes, and Seconds of the Equator into 
Hours, Minutes, fyc, of Time. 

Multiply the quantity by 4, and call the product of the seconds, 
thirds ; of the minutes, seconds ; and of the degrees, minutes. 
Exam. 1. Convert 83° ll 7 52" into time. 
83° 11' 52" 
4 



5 h. 32 m. 4?8 . 28'" 

2. Convert 34° 57' 46" into time. 

Ans. 2h. I9m. 51sec. 4"'. 



PROBLEM IV. 

To convert Hours, Minutes, and Seconds of Time into Degrees, 
Minutes, and Seconds of the Equator. 

Reduce the hours and minutes to minutes : divide by 4, and 
call the quotient of the minutes, degrees ; of the seconds, minutes ; 
and multiply the remainder by 15, for the seconds. 
Exam. 1. Convert 7h. 9m. 34sec. into degrees, &c. 
7 h - 9 m - 34 s * 
60 



4 ) 429 34 



107° 23' 30" 
2. Convert 1 lh. 24m. 45s. into degrees, &c. 

Ans. 171° 11' 16". 
35 



274 ASTRONOMICAL PROBLEMS. 



PROBLEM V. 

The Longitudes of two Places, and the Time at one of them 
being given, to find the corresponding Time at the other. 

When the given time is in the morning, change it to astronomi- 
cal time, by adding 12 hours, and diminishing the number of the 
day by a unit. When the given time is in the evening, it is al- 
ready in astronomical time. 

Find the difference of longitude of the two places, by taking the 
numerical difference of their longitudes, when these are of the 
same name, that is, both east or both west ; and the sum, when 
they are of different names, that is, one west and the other east. 
When one of the places is Greenwich, the longitude of the other 
is the difference of longitude. 

Then, if the place at which the time is required is to the east 
of the place at which the time is given, add the difference of longi- 
tude, in time, to the given time ; but, if it is to the west, subtract 
the difference of longitude from the given time. The sum or re- 
mainder will be the required time. 

Note. The longitudes used in the following examples, are given 
in Table I. 

Exam. 1. When it is October 25th, 3h. 13m. 22sec. A. M. at 
Greenwich, what is the time as reckoned at New York? 
Time at Greenwich, October, 24 d ' 15 h - 13 m - 22 s - 
Diff. of Long. ... 4 56 4 



Time at New York . . 24 10 17 18 P. M. 

2. When it is June 9th, 5h. 25m. lOsec. P. M. at Washington, 
what is the corresponding time at Greenwich ? 

Time at Washington, June, 9 d - 5 h - 25 m - 10 8 - 
Diff. of Long. ... 586 

Time at Greenwich . .9 10 33 16 P. M. 

3. When it is January 15th, 2h. 44m. 23sec. P. M. at Pans, 
what is the time at Philadelphia ? 

Longitude of Paris . . h - 9 m - 21 8 .6 E. 
Do. of Philadelphia, . 5 39 .6 W. 



Time at Paris, January, 
Diff. of Long. 

Time at Philadelphia, . 14 21 34 22 

Or January 15th, 9h. 34m. 22sec. A. M. 

4. When it is March 31st, 8h. 4m. 21 sec. P. M. at New Haven, 
what is the corresponding time at Berlin ? 

Ans. April 1st, lh. 49m. 43sec. A. M. 



5 


10 


1.2 


L5 C 


i. 2 h. 


44 m. 23'. 




5 


10 1 



TO CONVERT APPARENT INTO MEAN TIME. 275 

5. When it is August 10th, lOli. 32m. Msec. A. M. at Boston, 
what is the time at New Orleans ? 

Ans. Aug. 10th, 9h. 16m. 4sec. A. M. 

6. When it is noon of the 23d of December at Greenwich, what 
is the time at New York ? 

Ans, Dec, 23d, 7h. 3m. 55sec. A. M 



PROBLEM VL 

The Apparent Time being given, to find the corresponding Mean 
Time ; or the Mean Time being given to find the Apparent. 

When the given time is not for the meridian of Greenwich, re- 
duce it to that meridian by the last problem. Then find by the 
tables the sun's mean longitude corresponding to this time. Thus, 
from Table XVIII take out the longitude answering to the given 
year, and from Tables XIX, XX, and XXI, take out the motions 
in longitude for the given month, days, hours, and minutes, neg- 
lecting the seconds. The sum of the quantities taken from the 
tables, rejecting 12 signs, when it exceeds that quantity, will be 
the sun's mean longitude for the given time. 

With the sun's mean longitude thus found, take the Equation 
of Time from Table XII. Then, when Apparent Time is given 
to find the Mean, apply the equation with the sign it has in the 
table ; but when Mean Time is given to find the Apparent, apply 
it with the contrary sign ; the result will be the Mean or Apparent 
Time required. 

This rule will' be sufficiently exact for ordinary purposes, for 
several years before and after the year 1840, When the given 
date is a number of years distant from this epoch, take also with 
the sun's mean longitude the Secular Variation of the Equation of 
Time from Table XIII, and find by simple proportion the variation 
in the interval between the given year and 1840, The result, ap- 
plied to the equation of time taken from Table XII, according to 
its sign, if the given time is subsequent to the year 1840, but with 
the opposite sign if it is prior to 1840, will give the equation of 
time at the given date, which apply to the given time as above 
directed. 

Note 1. When the exact mean or apparent time to within a 
small fraction of a second is demanded, take the numbers in the 
columns entitled I, II, III, IV, V, N, in Tables, XVIII, XIX, 
XX, answering respectively to the year, month, days, and hours, 
of the given time. With the respective sums of the numbers 
taken from each column, as arguments, enter Table XIV, and 
take out the corresponding quantities. These quantities added to 
the equation of time as given by Tables XII and XIII, and the 



276 



ASTRONOMICAL PROBLEMS, 



constant 3.0s. subtracted, will give the true Equation of Time,- if 
the given time is Mean Time. When Apparent Time is given, it 
will be farther necessary to correct the equation of time as given 
by the tables, by stating the proportion, 24 hours : change of 
equation for 1° of longitude : : equation of time : correction. 

Note 2. The Equation* of Time is given in the Nautical Alma- 
nac for each day of the year, at apparent, and also at mean noon f 
on the meridian of Greenwich, and can easily be found for any 
intermediate time by a proportion. Direction* for applying it to the 
given time are placed at the head of the column. The Equation 
is given on the first and second pages of each month. 

Exam. 1. On the 16th of July, 1840, when it is 9h. 35m. 22s, 
P. M., mean time at New York, what is the apparent time at the 
same place ? 

Time at New York, July, 1840, 16 d - 9 b - 35 m - 22 8 - 
Diff. of Long. . 4 56 4 



Time at Greenwich, July, 1840, 



1840 

July 

16d. 

14h. 

31m, 



16 14 31 26 

M, Long. 

9 8 - 10° 12' 49" 

5 29 23 16 

14 47 5 

34 30 

1 16 



M. Long, . , , 3 24 58 56 

The equation of time in Table XII, corresponding to 3'- 24° 58^ 
56", is -r- 5"°- 44 s - 

Mean Time at New York, July, 1840, 16 d - 9 h - 35 m - 22 s - 
Equation of time, sign changed, , — 5 44 

Apparent Time, . , . , 16 9 29 38 P,M, 

2. On the 9th of May, 1842, when it is 4h. 15m. 21 sec. A. M, 
apparent time at New York, what is the mean time at the same 
place, and also at Greenwich ? 

Time at New York, May, 1842, 8 d - 16 b - 15 m - 21 9 - 
Diff. of Long. ... 4 56 4 



Time at 


Greenwich, 


. 




. 




M. Long. 




1842 


9 B 


10 c 


43' 


18" 


May 


3 


28 


16 40 


8d. 




6 


53 


58 


21h. 






51 


45 


11m. 








27 



8 21 11 25 



M. Long 



1 16 46 8. Equa. of time, — 3m. 45s, 



TO CONVERT APPARENT INTO MEAN TIME. 



277 



Apparent Time at Greenwich, May, 1 842, 
Equation of Time, ..... 

Mean Time at Greenwich, . 

Diff. of Long. ..... 



gd. 2lh- Urn. 25*. 

-3 45 

8 21 7 40 
4 56 4 

8 16 11 36 



Mean Time at New York, . 
Or, May 9th, 4h. 11m. 36s. A. M. 

3. On the 3d of February, 1855, when it is 2h. 43m. 36s. appa- 
rent time at Greenwich, what is the exact mean time at the same 
place ? 

Appar. Time at Greenwich, Feb., 1855, 3d. 2h. 43m. 36s. 



1855 . . 


M. Long. 


1 


II. 


m. 


rv. 


V. 


N. 


9 s 10° 34' 30" 


433 


279 


806 


889 


866 


863 


Feb. . . 


I 33 18 


47 


85 


1.38 


45 


7 


5 


3d. . . 


1 58 17 


68 


5 


9 


3 








2h. . . 


4 56 


3 












43m. . . 


i 46 














10 13 12 47 


551 


369 


953 


937 


873 


868 



Appar. Time at Greenwich, Feb., 1855, 3 d - 2 h - 43 m - 36 s - 
Equation of time by Table XII, . +14 8.6 

lOOyrs. : 13s. (Sec. Var., Table XII$ 

: : 15yrs.; 1.9s. . . —1.9 



Approx. Mean Time at Greenwich, 
24h. : 6s. (change of equa. for 1* 



of 



long.):; 14m. 
II. III. . 
II. IV. . 
II. V. 
L . 
N. . 
Constant . 



0,1s. 



3 2 57 42.7 

+0.1 
0.8 
0.4 
1.0 
0.3 
0.1 

—3.0 



Mean Time at Greenwich, 3 2 57 42.4 

4. On the 18th of November, 1841, when it is 2h. 12m. 26sec. 
A. M. mean time at Greenwich, what is the apparent time at 
Philadelphia ? Ans. Nov. 17th, 9h. 26m. 28s. P. M. 

5. On the 2d of February, 1839, when it is 6h. 32m. 35sec. 
P. M., apparent time at New Haven, what is the mean time at the 
same place ? Ans. 6h. 46m. 39s. P. M. 

6. On the 23d of September, 1850, when it is 9h. 10m. 12sec. 
mean time at Boston, what is the exact apparent time at the same 
place? Ans. 9h. 18m. 1.0s. 



278 ASTRONOMICAL PROBLEMS. 



PROBLEM VII. 



To correct the Observed Altitude of a Heavenly Body for Re- 
fraction. 

With the given altitude take the corresponding refraction from 
Table VIII. Subtract the refraction from the given altitude, and 
the result will be the true altitude of the body at the given station. 

This rule will give exact results if the barometer stands at 30 
inches, and Fahrenheit's thermometer at 50°, and results suffi- 
ciently exact for ordinary purposes in any state of the atmosphere. 
When there is occasion for greater precision, take from Table IX 
the corrections for + 1 inch in the height of the barometer, and 
■ — 1° in the height of Fahrenheit's thermometer, and compute the 
corrections for the difference between the observed height of the 
barometer and 30in. and for the difference between the observed 
height of the thermometer and 50°. Add these to the mean re- 
fraction taken from Table VIII, if the barometer stands higher 
than 30in. and the thermometer lower than 50° ; but in the oppo- 
site case subtract them, and the result will be the true refraction, 
which subtract from the observed altitude. 

Exam. 1. The observed altitude of the sun being 32° 10' 25", 
what is its true altitude at the place of observation ? 

Observed alt. . . . 32° 10' 25" 
Refraction (Table VIII) . — 1 32 



True alt. at the station, „ 32° 8 53 

2. The observed altitude of Sirius being 20° 42' 11", the ba- 
rometer 29.5 inches, and the thermometer of Fahrenheit 70°, 
required the true altitude at the place of observation. The differ- 
ence between 29.5 inches and 30 inches is 0.5 inches, and the 
difference between 70° and 50° is 20°. 
Obs. alt. . 20° 42' 11".0 



Refrac. (Table VIII), 2' 33".0; Bar.+lin.,5".12;ther.-l c \0".31G 
Corr.for— 0.5in.,bar. —2 .6 .5 20 

Corr.for+20°,ther. — 6 .2 

2.560 6.20 



True refrac. . 2 24 .2 



True alt. 20 39 46 .8 

3. The observed altitude of the moon on the 11th of April, 1838, 
being 14° 17' 20", required the true altitude at the place of obser- 
vation. Ans. 14° 13' 35". 

4. Let the observed altitude of Aldebaran be 48° 35' 52", the 
barometer at the same time standing at 30.7 inches, and the ther- 
mometer at 42°, required the true altitude. Ans. 48° 34' 58".8. 



TO DEDUCE THE TRUE FROM THE APPARENT ALTITUDE. 279 



PROBLEM VIII. 

The Apparent Altitude of a Heavenly Body being given, to find 
its True Altitude. 

Correct the observed altitude for refraction by the foregoing 
problem. Then, 

1. If the sun is the body whose altitude is taken,. find its paral- 
lax in altitude by Table X, and add it to the observed altitude cor- 
rected for refraction. The result will be the true altitude sought. 

2. If it is the altitude of the moon that is taken, and the hori- 
zontal parallax at the time of the observation is known, find the 
parallax in altitude by the following formula : 

log. sin (par. in alt.) = log. sin(hor.par.) -flog, cos (app.alt.) — 10 ; 

and add it, as before, to the apparent altitude corrected for refrac- 
tion. 

3. If one of the planets is the body observed, the following for- 
mula will serve for the determination of the parallax in altitude 
when the horizontal parallax is known : 

log. (par. in alt.) = log. (hor. par.) + log. cos (appar. alt) — 10. 

Note 1 . The equatorial horizontal parallax of the moon at any 
given time may be obtained from the tables appended to the work. 
(See Problem XIV.) But it can be had much more readily from 
the Nautical Almanac. The equatorial horizontal parallax being 
known, the horizontal parallax at any given latitude may be ob- 
tained by subtracting the Reduction of Parallax, to be found in 
Table LXIV. The horizontal parallax of any planet, the altitude 
of which is measured, may also be derived from the Nautical Al- 
manac. 

Note 2. The fixed stars have no sensible parallax, and thus the 
observed altitude of a star, corrected for refraction, will be its true 
altitude at the centre of the earth as well as at the station of the 
observer. 

Note 3. If the true altitude of a heavenly body is given, and it 
is required to find the apparent, the rules for finding the parallax 
in altitude and the refraction are the same as when the apparent 
altitude is given ; the true altitude being used in place of the ap- 
parent. But these corrections are to be applied with the opposite 
signs from those used in the determination of the true altitude from 
the apparent ; that is, the parallax is to be subtracted, and the re- 
fraction added. It will also be more accurate to make use of 
equa. (10), p. 52, in the case of the moon. 

Exam. 1. The observed altitude of the sun on the 1st of May, 
1837, being 26° 40' 20", what is its true altitude? 



280 



ASTRONOMICAL PROBLEMS. 



Obs. alt. 
Refraction . 

True alt. at the station, 
Parallax in alt. (Table X), 



26 y 40' 20" 
— 1 56 

26 38 24 

+ 8 

26 38 32 



True altitude 

2. Let the apparent altitude of the moon at New York on the 
17th of March, 1837, 8h. P. M., be 66° 10' 44" ; the barometer 
30.4in. and the thermometer 62° ; required the true altitude. 
Appar. alt. . . 66° 10' 44" 

Mean refrac. . 25.7 

Corr. for -f 0.4in., bar. + 0.3 

Corr. for + 12°, ther. —0.6 



True refrac. 



25.4 



True alt. at N.York, 66 10 18.6 
£qua. par. by N, Almanac, 54' 13" 
Reduc. for lat. 40°, 4 



Hor. par. at New York, 54 9 
Par. in alt. 



logarithms, 
cos. 9.60637 



sin. 8.19731 



21 52 



sin. 7.80368 



True altitude . . 66 32 11 

3. On the 18th of February, 1837, the true meridian altitude of 
the planet Jupiter at Greenwich was 56° 54' 57", what was its 
apparent altitude at the time of the meridian passage, the horizontal 
parallax being taken at 1".9, as given by the Nautical Almanac ? 

True alt. . . 56° 54' 57'' . cos. 9.7371 



Hor. par. 1".9 

Par. in alt. 
Refraction 

Appar. alt. 



log. 0.2787 



—1.0 
+ 37.9 



log. 0.0158 



56 55 34 

4. What will be the true altitude of the sun on the 22d of Sep- 
tember, 1840, at the time its apparent altitude is 39° 17' 50" ? 

Ans. 39° 16' 46". 

5. Given 29° 33' 30" the apparent altitude of the moon at Phil- 
adelphia on the 15th of June, 1837, at 9h. 30m. P.M., and 58' 33" 
the equatorial parallax of the moon at the same time, to find the 
true altitude. Ans. 30° 22' 41". 

6. Given 15° 24' 23" the true altitude of Venus, and 8" its hori- 
zontal parallax, to find the apparent altitude Ans. 15° 27' 41". 



TO FIND THE SUN's LONGITUDE, ETC , FROM TABLES. 281 



PROBLEM IX. 

To find the Surfs Longitude, Hourly Motion, and Semi-diameter, 
for a given time, from the Tables. 

For the Longitude. 

When the given time is not for the meridian of Greenwich, re- 
duce it to that meridian by Problem V ; and when it is apparent 
time, convert it into mean time by the last problem. 

With the mean time at Greenwich, take from Tables XVIII, 
XIX, XX, and XXI, the quantities corresponding to the year, 
month, day, hour, minute, and second, (omitting those in the last 
two columns,) and place them in separate columns headed as in 
Table XVIII, and take their sums.* The sum in the column enti- 
tled M. Long, will be the tabular mean longitude of the sun ; the 
sum in the column entitled Long. Perigee will be the tabular lon- 
gitude of the sun's perigee ; and the sums in the columns I, II, 
III, IV, V, N, will be the arguments for the small equations of the 
sun's longitude, including the equation of the equinoxes in longi- 
tude. 

Subtract the longitude of the perigee from the sun's mean longi- 
tude, adding 12 signs when necessary to render the subtraction 
possible ; the remainder will be the sun's mean anomaly. With 
the mean anomaly take the equation of the sun's centre from Ta- 
ble XXV, and correct it by estimation for the proportional part of 
the secular variation in the interval between the given year and 
1840 ; also with the arguments I, II, III, IV, V, take the corre- 
sponding equations from Tables XXVIII, XXX, XXXI, and 
XXXII. The equation of the centre and the four other equations, 
together with the constant 3", added to the mean longitude, will 
give the sun's True Longitude, reckoned from the Mean Equinox. 

"With the argument N take the equation of the equinoxes or Lu- 
nar Nutation in Longitude from Table XXVII. Also take the So- 
lar Xutation in longitude, answering to the given date, from the 
same table. Apply these equations according to their signs to the 
true longitude from the mean equinox, already found ; the result 
will be the True Longitude from the Apparent Equinox. 

For the Semi-diameter and Hourly Motion. 

With the sun's mean anomaly, take the hourly motion and semi- 
diameter from Tables XXIII and XXIV. 

* In adding quantities that are expressed in signs, degrees, &c, reject 12 or 24 
signs whenever the sum exceeds either of these quantities. In adding arguments 
expressed in 100 or 1000, &c. parts of the circle, when they consist of two figures, 
reject the hundreds from the sum ; when of three figures, the thousands ; and 
when of four figures, the ten thousands. 

36 



282 



ASTRONOMICAL PROBLEMS. 



Notes. 

1 . If the tenths of seconds be omitted in taking the equations 
from the tables of double entry, the error cannot exceed 2" ; in 
case the precaution is taken to add a unit, whenever the tenths ex- 
ceed .5. 

2. The longitude of the sun, obtained by the foregoing rule, 
may differ about 3" from the same as derived from the most accu- 
rate solar tables now in use. When there is occasion for greater 
precision, take from Tables XVIII, XIX, and XX, the quantities 
in the columns entitled VI and VII, along with those in the other 
columns. With the sums in these columns, and those in the col- 
umns I, II, as arguments, take the corresponding equations from 
Tables XXIX and XXXIII. Also with the sun's mean anomaly 
take the equation for the variable part of the aberration from Ta- 
ble XXXIV. Add these three equations along with the others to 
the mean longitude, and omit the addition of the .constant 3". The 
result will be exact to within a fraction of a second. 

Exam. 1. Required the sun's longitude, hourly motion, and se- 
mi-diameter, on the 25th October, 1837, at llh. 27m. 38s. A. M. 
mean time at New York. 

Mean time at N. York, Oct. 1837, 24 d - 23 h - 27 m - 38 s - 
Diff. of Long 4 56 4 



Mean time at Greenwich, 



25 4 23 42 



1837 . 
October 
25d. . 
4h. . 
23m. . 
42s. . 



Eq. Sun's Cent 
I. . . 
II. III. . . 
II. IV. . . 
II. V. . . 
Const. . . . 



Lunar Nutation 
Solar Nutation 

Sun'strue long. 



M. Long. 


S 


/ // 


9 10 


55 47.2 


8 29 


4 54.1 


23 


39 19.9 




9 51.4 




56.7 
1.7 




7 3 
11 28 


50 51.0 
12 43.5 




2.5 




9.0 




7.7 




19.3 




3.0 


7 2 


4 16.0 




— 6.3 




— 1.2 


7 2 


4 8.5 



Long. Perigee. I. II 



9 10 8 5 816 280 
46 250 748 
4810! 66 

6i 



9 10 8 55 882 94 872 753 416 939 
7 3 50 5l| 



III. IT 



549 

215 

107 

1 



321 

397 

35 



V. 



348 

63 

5 



N. 



895 

40 

4 



9 23 41 56 Mean Anomaly. 
Sun's Hourly Motion, . . 2' 29". 7 
Sun's Semi-diameter . 16' 17".2 



2. Required the sun's longitude, hourly motion, and semi-diam- 
eter, on the 15th of July, 1837, at 8h. 20m. 40s. P. M. mean time 
at Greenwich. 



TO FIND THE APPARENT OBLIQUITY OF THE ECLIPTIC. 



283 





M. Long. 


Long. Peri. 


I. 


II. 


IIIIV. 


V. 


N. 


VI. 


VII. 


so' " | s ° / " 






1 










1837 . . . 


9 10 55 47.2 9 10 8 5 


816 


280 


549 321348 


895 


787 


600 


July . . . 


5 28 24 7.8 


31 


129496 


806;263 41 


27 


569 


17 


15d. . . . 


13 47 56.6 


2 


473 


38 


62 


20 


3 


2 


508 


2 


8h. ... 


19 42.8 




11 


1 


1 








11 




20m. . . . 
40s 


49 3 




















l!6'9 10 8 38 


429 


815 


418 


604392 


924 


875 


619 




lo OQ OQ OC 






3 23 28 25.3 
11 29 33 10.3 






Eq. Sun's Cent. 


6 13 19 47 Mean Anomaly. 


I. . . 


10.7 




II. III. . • 


6.6 


Sun's Hourly Motion, . . 2' 23". 1 


|II. IV. . . 


5.0 




II. V. . . 


7.7 


Sun's Semi-diameter, . . 15' 45".4 


i I. VI. . . 


1.8 




|II. VII. . . 


0.2 




|Aber. . . . 

1 

! 


0.6 




3 23 2 8.2 


Lunar Nutation 


— 7.8 




Solar Nutation 
Sun's true long. 


+ 0.8 




3 23 2 1.2 





















3. Required the sun's longitude, hourly motion, and semi-diam- 
eter, on the 10th of June, 1838, at 9h. 45m. 26s. A. M. mean time 
at Philadelphia, (omitting the three smallest equations of longi- 
tude.) 

Ans. Sun's longitude, 2 s - 19° 11/ 57" ; hourly motion, 2' 23".3 ; 
semi-diameter, 15' 46". 1. 

4. Required the sun's longitude, hourly motion, and semi-diam- 
eter, on the 1st of February, 1837, at 12h. 30m. 15s. mean astro- 
nomical time at Greenwich. 

Ans. Sun's longitude, 10 s - 13° V 44". 6 ; hourly motion, 2' 
32".l ; semi-diameter, 16' 14".7. 



PROBLEM X. 



To find the Apparent Obliquity of the Ecliptic, for a given time, 
from the Tables. 

t Take the mean obliquity for the given year from Table XXII. 
Then with the argument N, found as in the foregoing problem, 
and the given date, take from Table XXVII the lunar and solar 
nutations of obliquity. Apply these according to their signs to the 
mean obliquity, and the result will be the apparent obliquity. 

Exam. 1 . Required the apparent obliquity of the ecliptic* on the 
15th of March, 1839. 



284 ASTRONOMICAL PROBLEMS- 



1839, . 


3 












March, 


9 












15d. . 


2 
















M. 


Obliquity, 


23° 


27' 36" 


.9 




14 


. 


. 




+ 9 


.1 


Solar Nutation for March 15th, 




+ 


.5 



Apparent Obliquity, . . 23 27 46 .5 
2. Required the apparent obliquity of the ecliptic on the 12th 
of July, 1845. Ans. 23° 27' 28 ,/ .2. 

PROBLEM XI. 

Given the Surfs Longitude and the Obliquity of the Ecliptic, to. 
find his Right Ascension and Declination* 

Let w = obliquity of the ecliptic ; L = sun's longitude ; R = 
sun's right ascension ; and D = sun's declination ; then to find R 
and D, we have 

log. tang R = log. tang L + log. cos u — 10, 
log. sin D = log. sin L + log. sin « — 10. 

The right ascension must always be taken in the same quadrant 
as the longitude. The declination must be taken less than 90° ; 
and it will be north or south according as its trigonometrical sine 
comes out positive or negative. 

Note. The sun's right ascension and declination are given in 
the Nautical Almanac for each day in the year at noon on the me- 
ridian of Greenwich, and may be found at any intermediate time 
by a proportion. 

Exam. 1 . Given the sun's longitude 205° 23' 50", and the ob- 
liquity of the ecliptic 23° 27' 36", to find his right ascension and 
declination. 

L=205° 23' 50" ... tan. 9.67649 
w = 23 27 36 . . . cos. 9.96253 



R = 203 32 5 . . . tan. 9.63902 



L= 205 23 50 . . . sin. 9.63235— 
w = 23 27 36 . . . sin. 9.60000 



D= 9 49 52 S. . . . sin. 9.23235- 
2. The obliquity of the ecliptic being 23° 27' 30", required 

* The obliquity of the ecliptic at any given time for which the sun's longitude 
is known, is found by the foregoing Problem. 



285 

the sun's right ascension and declination when his longitude is 
44° 18' 25". 

Ans. Right ascension 41° 5(y 30", and declination 16°8'40"N, 



PROBLEM XII. 

Given the Sun's Right Ascension and the Obliquity of the Eclip 
tic, to find his Longitude and Declination. 

Using the same notation as in the last problem, we have, to find 
the longitude and declination, 

log. tang L •= log. tang R + ar. co. log. cos w, 
log. tang D = log. sin R + log. tang u — 10. 

Exam. 1. What is the longitude and declination of the sun, 
when his right ascension is 142° 11' 34", and the obliquity of the 
ecliptic 23° 27' 40" ? 

R = 142° 11' 34" . . . tan. 9.88979- 
w = 23 27 40 . . ar. co. cos. 0.03747 



L = 139 46 30 . . . tan. 9.92726 — 



R= 142 11 34 . . . sin. 9.78746 
w= 23 27 40 . . . tan. 9.63750 



D= 14 53 55N... . . tan. 9.42496 
2. Given the sun's right ascension 310° 25' 11", and the obli- 
quity of the ecliptic 23° 27' 35", to find the longitude and declina- 
tion. 

Ans. Longitude 307° 59' 57", and declination 18° 17' 0" S* 



PROBLEM XIIL 

The Sun's Longitude and the Obliquity of the Ecliptic being 
given, to find the Angle of Position. 

Let p = angle of position ; w = obliquity of the ecliptic ; and 
L = sun's longitude. Then, 

log. tangp = log. cos L -f- log. tang w — 10. 

The angle of position is always less than 90°. The northern 
part of the circle of latitude will lie on the west or east side of the 
northern part of the circle of declination, according as the sign of 
the tangent of the angle of position is positive or negative. 

Exam. 1. Given the sun's longitude 24° 15' 20", and the obli- 
quity of the ecliptic 23° 27' 32", required the angle of position. 



286 ASTRONOMICAL PROBLEMS. 

L = 24° 15' 20" . . cos. 9.95980 
w = 23 27 32 . . tan. 9.63745 



p= 21 35 10 . . tan. 9.59731 

The northern part of the circle of latitude is to the west of the 
circle of declination. 

I 2. When the sun's longitude is 120° 18' 55", and the obliquity 
of the ecliptic 23° 27' 30", what is the angle of position ? 

Ans. 12° 21' 17" ;. and the northern part of the circle of latitude 
lies to the east of the circle of declination. 



PROBLEM XIV. 

To find from the Tables, the Moon's Longitude, Latitude, Equa- 
torial Parallax, Semi-diameter, and Hourly Motion in longi- 
tude and Latitude, for a given time. 

When the given time is not for the meridian of Greenwich, re- 
duce it to that meridian, and when it is apparent time convert it 
into mean time. 

Take from Table XXXV, and the following tables, the argu- 
ments numbered 1, 2, 3, &c, to 20, for the given year, and their 
variations for the given month, days, &c, and find the sums of the 
numbers for the different arguments respectively ; rejecting the 
hundred thousands and also the units in the first, the ten thousands 
in the next eight, and the thousands in the others. The resulting 
quantities will be the arguments for the first twenty equations of 
longitude. 

With the same time, take from the same tables the remaining 
arguments with their variations, entitled Evection, Anomaly, Va- 
riation, Longitude, Supplement of the Node, II, V, VI, VII, VIII, 
IX, and X ; and add the quantities in the column for the Supple- 
ment of the Node. 

For the Longitude. 

With the first twenty arguments of longitude, take from Tables 
XLI to XLVI, inclusive, the corresponding equations ; and with 
the Supplement of the Node for another argument, take the corre- 
sponding equation from Table XLIX. Place these twenty-one 
equations in a single column, entitled Eqs. of Long. ; and write 
beneath them the constant 55". Find the sum of the whole, and 
place it in the column of Evection. Then the sum of the quanti- 
ties in this column will be the corrected argument of Evection. 

With the corrected argument of Evection, take the Evection 
from Table L, and add it to the sum in the column of Eqs. of 
Long. Place this in the column of Anomaly. Then the sum of 
the quantities in this column will be the corrected Anomaly. 



TO FIND THE MOON'S LONGITUDE, ETC. 287 

With the corrected Anomaly, take the Equation of the Centre 
from Table LI, and add it to the last sum in the column of Eqs. 
of Long. Place the resulting sum in the column of Variation. 
Then the sum of the quantities in this column will be the corrected 
argument of Variation. 

With the corrected argument of Variation, take the variation 
from Table LII, and add it to the last sum in the column of Eqs. 
of Long. ; the result will be the sum of the principal equations 
of the Orbit Longitude, amounting in all to twenty- four, and the 
constants subtracted for the other equations. Place this sum in 
the column of Longitude. Then the sum of the quantities in this 
column will be the Orbit Longitude of the Moon, reckoned from 
the mean equinox. 

Add the orbit longitude to the supplement of the node, and the 
resulting sum will be the argument of Reduction. 

With the argument of Reduction, take the Reduction from Ta- 
ble LIII, and add it to the Orbit Longitude. The sum will be the 
Longitude as reckoned from the mean equinox. With the Supple- 
ment of the Node, take the Nutation in Longitude from Table 
LIV, and apply it, according to its sign, to the longitude from the 
mean equinox. The result will be the Moon's True Longitude 
from the Apparent Equinox. 

For the Latitude. 

The argument of the Reduction is also the 1st argument of Lat- 
itude. Place the sum of the first twenty-four equations of Longi- 
tude, taken to the nearest minute, in the column of Arg. II. Find 
the sum of the quantities in this column, and it will be the Arg. II 
of Latitude, corrected. The Moon's true Longitude is the 3d ar- 
gument of Latitude.. The 20th argument of Longitude is the 4th 
argument of Latitude. Take from Table LVIII the thousandth 
parts of the circle, answering to the degrees and minutes in the 
sum of the first twenty-four equations of longitude, and place it in 
the columns V, VI, VII, VIII, and IX ; but not in the column X. 
Then the sums of the quantities in columns V, VI, VII, VIII, IX, 
and X, rejecting the thousands, will be the 5th, 6th, 7th, 8th, 9th, 
and 10th arguments of Latitude. 

With the Arg. I of Latitude, take the moon's distance from the 
North Pole of the Ecliptic, from Table LV ; and with the remain- 
ing nine arguments of latitude, take the corresponding equations 
from Tables LVI, LVII, and LIX. The sum of these quantities, 
increased by 10", will be the moon's true distance from the North 
Pole of the Ecliptic. The difference between this distance and 
90° will be the Moon's true Latitude ; which will be North or 
South, according as the distance is less or greater than 90°. 

For the Equatorial Parallax. 
With the corrected arguments, Evection, Anomaly, and Varia- 



288 ASTRONOMICAL PROBLEMS. 

tion, take out the corresponding quantities from Tables LXI, 
LXII, and LXIII. Their sum, increased by 7", will be the Equa- 
torial Parallax. 

For the Semi-diameter. 

With the Equatorial Parallax as an argument, take out the 
moon's semi-diameter from Table LXV. 

For the Hourly Motion in Longitude. 

With the arguments 2, 3, 4, 5, and 6 of Longitude, rejecting the 
two right-hand figures in each, take the* corresponding equations 
of the hourly motion in longitude from Table LXVII. Find the 
sum of these equations and the constant 3", and with this sum at 
the top, and the corrected argument of the Evection at the side, 
take the corresponding equation from Table LXIX ; also with the 
corrected argument of the Evection take the corresponding equa- 
tion from Table LXVIII. 

Add these equations to the sum just found, and with the result- 
ing sum at the top, and the corrected anomaly at the side, take the 
corresponding equation from Table LXX ; also with the corrected 
anomaly take the corresponding equation from Table LXXI. 

Add these two equations to the sum last found, and with the re- 
sulting sum at the top, and the corrected argument of the Variation 
at the side, take the corresponding equation from Table LXXII. 
With the corrected argument of the Variation, take the correspond- 
ing equation from Table LXXIII. 

Add these two equations to the sum last found, and with the re- 
sulting sum at the top, and the argument of the Reduction at the 
side, take the corresponding equation from Table LXXIV. Also, 
with the argument of the Reduction take the corresponding equa- 
tion from Table LXXV. These two equations, added to the last 
sum, will give the sum of the principal equations of the hourly 
motion in longitude, and the constants subtracted for the others. 
To this add the constant 27' 24". 0, and the result will be the 
Moon's Hourly Motion in Longitude. 

For the Hourly Motion in Latitude. 

With the argument I of Latitude, take the corresponding equa- 
tion from Table LXXIX. With this equation at the side, and the 
sum of all the equations of the hourly motion in longitude, except 
the last two, at the top, take the corresponding equation from Ta- 
ble LXXXI. With the argument II of Latitude, take the corre- 
sponding equation from Table LXXXII. And with this equation 
at the side, and the sum of all the equations of the hourly motion 
in longitude, except the last two, at the top, take the equation from 
Table LXXXIII. Find the sum of these four equations and the 



TO FIND THE MOON'S LONGITUDE, ETC. 289 

constant 1". To the resulting sum apply the constant — 237".2. 
The difference will be the Moon's true Hourly Motion in Latitude. 
The moon will be tending North or South, according as the sign 
is positive or negative. 

Note. The errors of the results obtained by the foregoing rules, 
occasioned by the neglect of the smaller equations, cannot exceed 
for the longitude 15", for the latitude 8", for the parallax 7", for 
the hourly motion in longitude 5", and for the hourly motion in 
latitude 3" ; and they will generally be very much less. When 
greater accuracy is required, take from Tables XXXV to XXXIX 
the arguments from 21 to 31, along with those from 1 to 20, and 
their variations. The sums of the numbers for these different ar- 
guments, respectively, will be the arguments of eleven small addi- 
tional equations of longitude. Also, take from the same tables the 
arguments entitled XI and XII, along with those in the preceding 
columns. Retain the right-hand figure of the sum in column 1 of 
arguments, and conceive a cipher to be annexed to each number 
in the columns of arguments of Table XLI. The numbers in the 
columns entitled Diff.for 10, will then be the differences for a va- 
riation of 100 in the argument. 

For the Longitude. With the arguments 21 to 31, take the cor- 
responding equations from Tables XLVII and XLVIII, and place 
them in the same column with the equations taken out with the 
arguments 1, 2, &c. to 20. Take also equation 32 from Table 
XLIX, as before. Find the sum of the whole, (omitting the con- 
stant 55",) and then continue on as above. The longitude from 
the mean equinox being found, take the lunar nutation in longitude 
from Table LIV, and the solar nutation answering to the given 
date from Table XXVII. Apply them both, according to their 
sign, to the longitude from the mean equinox, and the result will 
be the more exact longitude from the apparent equinox, required. 

For the Latitude. With the arguments XI and XII, take the 
corresponding equations from Table LIX. Add these with the 
other equations, and omit the constant 10". The difference be- 
tween the sum and 90° will be the more exact latitude. 

For the Equatorial Parallax. With the arguments 1, 2, 4, 5, 
6, 8, 9, 12, 13, take the corresponding equations from Table LX. 
Find the sum of these and the other equations, omitting the con- 
stant 7", and it will be the more exact value of the Parallax. 

For the Hourly Motion in Longitude. With the arguments 1, 
7; 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, and 18, of longitude, along 
with the arguments 2, 3, 4. 5, and 6, heretofore used, take the cor- 
responding equations froui Table LXVII. Find the sum of the 

37 



290 ASTRONOMICAL PROBLEMS. 

whole, omitting the constant 3", and proceed as in the rule already 
given. 

To obtain the motion in longitude for the hour which precedes 
or follows the given time, with the arguments of Tables LXX, 
LXXII, and LXXIV, take the equations from Tables LXXVI 
and LXXVII. Also, with the arguments of Evection, Anomaly, 
Variation, and Reduction, take the equations from Table LXXVIII. 
Find the sum of all these equations. Then, for the hour which fol- 
lows the given time, add this sum to the hourly motion at the given 
time already found, and subtract 2".0 ; for the hour which pre- 
cedes, subtract it from the same quantity, and add 2".0. 

It will expedite the calculation to take the equations of the sec- 
ond order from the tables at the same time with those of the first 
order which have the same arguments. 

For the Hourly Motion in Latitude. The moon's hourly mo- 
tion in latitude may be had more exactly by taking with the argu- 
ments of Latitude V, VI, &c. to XII, the corresponding equations 
from Table LXXX, and finding the sum of these and the other 
equations of the hourly motion in latitude. 

To obtain the moon's motion in latitude for the hour which pre- 
cedes or follows the given time, with the Argument I of Latitude, 
take the equation from Table LXXXIV, and with this equation 
and the sum of all the equations of the hourly motion in longitude 
except the last two, take the equation from Table LXXXV. Find 
the sum of these two equations. Then, for the hour which follows 
the given time, add this sum to the Hourly Motion in Latitude al- 
ready found, taken with its sign, and subtract 1".3; and for the 
hour which precedes, subtract it from the same quantity, and add 
1".3. 

It will also be more exact to enter Table LXXXI with the sum 
of all the equations of Tables LXXIX and LXXX, diminished 
by 1", instead of the equation of Table LXXIX, for the argument 
at the side. The numbers over the tops of the columns in Table 
LXXXI are the common differences of the consecutive numbers 
in the columns. The numbers in the last column are the common 
differences of the consecutive numbers in the same horizontal line. 

Exam. 1. Required the moon's longitude, latitude, equatorial 
parallax, semi-diameter, and hourly motions in longitude and lati- 
tude, on the 14th of October, 1838, at 6h. 54m. 34s. P. M. mean 
lime at New York. 

Mean time at New York, October, 14 d - 6 h - 54 m - 34 s - 
Diff. of Long 4 56 4 

Mean time at Greenwich, October, 14 11 50 38 



TO FIND THE MOON S LONGITUDE, ETC. 



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TO FIND THE MOON's REDUCED PARALLAX, ETC. 295 

Exam. 2. Required the moon's longitude, latitude, equatorial 
parallax, semi-diameter, and hourly motions in longitude and lati- 
tude, on the 9th of April, 1838, at 8h. 58m. 19s. P. M. mean time 
at Washington. 

Ans. Long. 6 s - 19° 45' 31".2; lat. 36' 21 // .9 S. ; equat. par. 
54' 36".3 ; semi-diameter 14' 52" .7 ; hor. mot. in longf. 30' 15".2 ; 
and hor. mot. in lat 2' 47". 0, tending south.* 



PROBLEM XV. 

The Moon's Equatorial Parallax, and the Latitude of a Place, 
being given, to find the Reduced Parallax and Latitude. 

With the latitude of the place, take the reductions from Table 
LXIV, and subtract them from the Parallax and Latitude. 

Exam. 1. Given the equatorial parallax 55' 15", and the lati- 
tude of New York 40° 42' 40" N., to find the reduced parallax and 
latitude. 

Equatorial parallax, ... 55' 15" 
Reduction, ..... 5 



Reduced parallax, . . . 55 10 

Latitude of New York, . 40° 42' 40" N. 

Reduction, . . . , 11.20 



Reduced Lat. of New York, 40 31 20 

2. Given the equatorial parallax 60' 36", and the latitude of 
Baltimore 39° 17' 23" N,, to find the reduced parallax and latitude. 

Ans. Reduced par. 60' 32", and reduced lat. 39° 6' 9". 

3. Given the equatorial parallax 57' 22", and the latitude of 
New Orleans 29° 57' 45" N., to find the reduced parallax and lat- 
itude. 

Ans. Reduced par. 57' 19", and reduced lat. 29° 47' 50". 



PROBLEM XVI. 

To find the Longitude and Altitude of the Nonagesimal Degree 
of the Ecliptic, for a given time and place. 

For the given time, reduced to mean time at Greenwich, find the 
sun's mean longitude and the argument N from Tables XVIII, 
XIX, XX, and XXI. To the sun's mean longitude, apply accord- 
ing to its sign the nutation in right ascension, taken from Table 

* The smaller equations were omitted in working this example. 



296 ASTRONOMICAL PROBLEMS. 

XXVII with argument N ; and the result will be the right ascen- 
sion of the mean sun, (see Art. 45,) reckoned from the true equi- 
nox. 

Reduce the mean time of day at the given place, expressed as 
tronomically, to degrees, &c, and add it to the right ascension of 
the mean sun from the true equinox. The sum, rejecting 360°, 
when it exceeds that quantity, will be the right ascension of the 
midheaven, or the sidereal time in degrees. 

Next, find the reduced latitude of the place by Problem XV ; 
and when it is north, subtract it from 90° ; but when it is south, 
add it to 90°. The sum or difference will be the reduced distance 
of the place from the north pole. 

Also, take the obliquity of the ecliptic for the given year from 
Table XXII.* 

These three quantities having been found, the longitude and alti- 
tude of the nonagesimal degree may be computed from the follow- 
ing formulae : 

log. cos \ (H — w) — log. cos \ (H + w) = A . . . (1); 

log. tang i (H - u) + 10 - log. tang i (H + w) = B . . . (2) ; 

log. tang E = A + log. tang \ (S — 90°) . . . (3) ; 

log. tang F = log. tang E + B . . . (4) ; 

N = E+F + 90° . . . (5); 

log. tang \h = log. cos E + log. tang 4- (H + w ) + ar. co. log. 

« cos F - 20 . . . (6). 

in which 

H = the reduced distance of the place from the north pole ; 

a) = the Obliquity of the Ecliptic ; 

S = the Sidereal Time converted into degrees ; 

N = the required Longitude of the Nonagesimal ; 

h = the required Altitude of the Nonagesimal ; 

E and F are auxiliary angles. 

We first find the logarithmic sums A and B. With these we de- 
termine the angles E and F by formulae (3) and (4), and with these 
again N and h by formulae (5) and (6). 

The angles E, F, are to be taken less than 180°; and less or 
greater than 90°, according as the sign of their tangent proves to be 
positive or negative. 

Note 1 . In case the given place lies within the arctic circle, we 
must take, in place of formula (5), the following : 
N = E -F + 90 . 



* If great precision is required, the apparent obliquity is to be used in place of the 
mean. (See Prob. X.) 



TO FIND THE LONG. AND ALT. OF NONAGESIMAL DEGREE. 297 



Note 2. As the obliquity of the ecliptic varies but slowly from 
year to year, the values which have once been found for the loga- 
rithms A, B, and log. tang £ (H + w) (C), will answer for several 
years from the date of their determination, unless very great accu- 
racy is required. 

Note 3. The angle h derived from formula (6), is the dis- 
tance of the zenith of the given place from the north pole of the 
ecliptic. This is not always equal to the altitude of the nonagesi- 
mal. Throughout the southern hemisphere, and frequently in the 
northern near the equator, it is the supplement of the altitude. In 
employing this angle in the following Problem, it is, however, for the 
sake of simplicity, called the altitude of the nonagesimal in all cases. 

Exam. 1 . Required the longitude and altitude of the nonagesi- 
mal degree of the ecliptic at New York, on the 18th of September, 
1838, at 3h. 52m. 56s. P. M. mean time. 

The sun's mean longitude taken from the tables, for the given 
time, is 5 s - 27° 19' 17", and the argument N is 987. The nutation 
taken from Table XXVII with argument N is — 1". Hence, the 
right ascension of the mean sun, reckoned from the true equinox, 
is 5 s - 27° 19' 16". The given time of day, expressed astronomi- 
cally, is 3h. 52m. 56sec. ; which in degrees is 58° 14' 0". 

•The reduced latitude of New York, found by Problem XV, is 
40° 31' 20", and this taken from 90° leaves the polar distance 49° 
28' 40". The obliquity of the ecliptic, derived from Table XXII, 
is 23° 27' 37". 

Given time in degrees, . . . 58° 14' 0" 
R. Asc. of mean sun, . . . 177 19 16 



Sidereal time in degrees (S), 



235 
90 



33 16 



2)145 33 16 



H . . 49° 28' 40" 
u . . 23 27 37 


i (S - 90) 

cos. 9.98870 . 
cos. 9.90535 . 


72 

tan. 
tan. 

cos. 
ar. c 

4" . 
8 


46 38 


Diff . . 26 1 3 
Sum . . 72 56 17 




\ diff. . . 13 31 . 
| sum . . 36 28 8 


+ 10,19.36366 
C. 9.86871 


i(S -90°) 72 46 38 


A. 0.08335 
. tan. 0.50866 


. B. 9.49495 


E . 75 38 55 

F . 50 41 55 
90 


. tan. 0.59201 . 

B. 9.49495 
. tan 0.08696 . 

£ alt. non. 16° 7' 5 


9.39422 

C. 9.86871 

o. cos. 0.19832 


tan. 9.46125 


long. non. 216 20 50 


alt. non. 32 15 4 
38 





298 ASTRONOMICAL PROBLEMS. 

2. Required the longitude and altitude of the nonagesimal de- 
gree of the ecliptic at New York, on the 10th of May, 1838, at 
llh. 33m. 56sec. P. M. mean time. 

Ans. Long. 200° 12' 23", and alt. 37° 0' 34". 

PROBLEM XVII. 

To find the Apparent Longitude and Latitude, as affected by 
Parallax, and the Augmented Semi-diameter of the Moon ; the 
Moon's True Longitude, Latitude, Horizontal Semi-diameter, 
and Equatorial Parallax, and the Longitude and Altitude of 
the Nonagesimal Degree of the Ecliptic, being given. 

We have for the resolution of this Problem the following for- 
mulae : 

log. x = log. P+ log. cos A+ar. co.log.cos X— 10 . . . (1); 

c = log. x + log. tang h — 10 ... (2) ; 
log. u = c + log. sin K — 10 . . . (3) ; 
log. w' = c+log. sin (K + u) — 10 . . . (4) ; 
log. p = c + log. sin (K + W) — 10 . . . (5) ; 
Appar. long. = true long. +p . . . (6) ; 

log. tang X' = log. p + ar. co. log. cos X + ar. co. log. u -f- log. 
sin(X-tf)- 10* . . . (7); 
log. v = log. P + log. cos h -+- log. cos X' — 10 ... (8); 
log. z = log. v + log. tang h + log. tang X' + log. cos 
(K-Hp)-30 . . . (9); 
ne= v — z . . . (10); 
Appar. lat. = true lat. — * . . . (11); 

log.R' = log. p -f- ar. co. log. cos X -f* ar. co. log. u + log. 
cosX' +log. R- 10 . . . (12): 
in which 

P == the Reduced Parallax of the Moon ; 
h = the Altitude of the Nonagesimal ; 
X = the True Latitude of the Moon (minus when south) ; 
K = the Longitude of the Moon, minus the longitude of the No- 
nagesimal ; 
p = the required Parallax in Longitude ; 
X ; = the approximate Apparent Latitude of the Moon ; 
* = the required Parallax in Latitude ; 
R == the True Semi-diameter of the Moon ; 
R ; = the Augmented Semi-diameter of the Moon ; 
x, u, u', v, z, are auxiliary arcs. 

* Formula (7) will be rendered more accurate by adding to it the ar. co. cos 
x — 10, and will generally give the apparent latitude with sufficient accuracy; 
thus rendering formulae (8), (9), (10), and (11) unnecessary. 



299 

Formulae (1), (2), (3), (4), and (5), being resolved in succession, 
we derive the apparent longitude from formula (6) ; then the appa- 
rent latitude from equations (7), (8), (9), (10), (11); and lastly, 
the augmented semi-diameter from equation (12.) 

The latitude of the moon must be affected with the negative 
sign when south ; and the apparent latitude will be south when it 
comes out negative. In performing the operations, it is to be re- 
membered that the cosine of a negative arc has the same sign as 
the cosine of a positive arc of an equal number of degrees ; but 
that the sine or tangent of a negative arc has the opposite sign from 
the sine or tangent of an equal positive arc. Attention must also 
be paid to the signs in the addition and subtraction of arcs. Thus, 
two arcs affected with essential signs, which are to be added to 
each other, are to be added arithmetically when they have like 
signs, but subtracted if they have unlike signs ; and when one arc 
is to be taken from another, its sign is to be changed, and the two 
united according to their signs. An arithmetical sum, when taken, 
will have the same sign as each of the arcs : and an arithmetical 
difference the same sign as the greater arc. 

The use of negative arcs may be avoided, though the calculation 
would be somewhat longer, by using the true polar distance d, and 
the approximate apparent polar distance d', in place of X and X', 
substituting sin d for cos X, cos (d + x) for sin (X — x), sin d' for 
cos X', log. co-tang d! for log. tang X' ; and observing that p is 
to be subtracted from the true longitude in case the longitude of 
the nonagesimal exceeds the longitude of the moon ; that z, when 
it -comes out negative, is to be added to v, which is always positive 
to the north of the tropic, otherwise subtracted ; and that the par- 
allax in latitude is to be applied according to its sign to the true 
polar distance. 

In seeking for the logarithms of the trigonometrical lines, it will 
be sufficient to take those answering to the nearest tens of seconds. 

Note 1 . When great accuracy is not desired, u! may be taken 
for p, from which it can never differ more than a fraction of a 
second. 

Note 2. In solar eclipses the moon's latitude is very small, and 
formula (7) may be changed into the following : 

log. X' = log. p + ar. co. log. cos X -f- ar. co. log. u -Hog. (X — x) — 1 

and cos X' omitted in formula (12) without material error. 

Formulae (8), (9), (10), and (11), may also now be dispensed 
with, unless very great precision is desired, and the value of X' 
given by the above formula taken for the apparent latitude. 

It is to be observed also, that in eclipses of the sun P is taken 
equal to the reduced parallax of the moon minus the sun's horizon- 
tal parallax. By this the parallax of the sun in longitude and lati- 
tude is referred to the moon, and the relative apparent places of 
the sun and moon are correctly obtained, without the necessity of 



SOO" ASTRONOMICAL PROBLEMS. 

a separate computation of the sun's parallax in longitude and 
latitude. 

Exam. 1 . About the time of the middle of the oecultation of the 
star Antares, on the 10th of May, 1838, the moon's longitude, by 
the Connaissance des Terns, was 247° 37' 6".7 ;. latitude 4° 14' 
14".7 S.; semi-diameter 15' 24". 2; and equatorial parallax 56' 
31".7 ' r and the longitude of the nonagesimal at New York wa& 
200° 12' 23" ; the altitude 37° 0' 34" ; required the apparent lon- 
gitude and latitude, and the augmented semi-diameter of the moon f 
at New Yorl&, at the time in question. 

Equatpar. 56' 31 ".7 Moon's long. 247° 37' 7" 

Reduction 4 .6 Long- nonag. 200 12 23 



F 

F 

h 


— f 
•- 

•• 
«- 

r. 

.- 


>6 27 . j 

3387".! 
. 87° 0'34" 

- 4 14 15 

45 12 . 

* 37 34 . 

* 47 24 44 . 

25 5 . 

. 47 49 49 . 
25 15 . 

. 47 49 59 . 

25 15.3 . 
, 247 37 6.7 


K 

h 

ar. co 
2712" . 

1505" . 
H515T.Z . 

I515".3 . 

i *' * 

, ar. co 
, ar. co 

• *■ 


= 47 24 44 

=37 34 

= -4 14 14.7 

log. 3.52983 

eos. 9.90230 


X 


a. 3.4321 & 
.cos-. 0.00119* 


X 

h 


log. 3.43332 
tan, &8772& 


K . 


€. 3.31 05T 
sin. 9.86701 


u 
K + tt 


log. 3.1775$ 

c. 3.31057 
sin. 9.86991 


«' 
K + uf 


log. 3.1804$ 

e. 3,31057 
sin. 9.86993 


True long. 


log. 3.18050* 


Appar. long 

p 

x— a?' . 

X 

u 


.. 248 2 22.0 
. -4 59 27 . 

• -& I 10 . 


fog. 3.18050' 
sin. 8.93957- 
. eos. 0.00119 
, log. 6.82242 

tan. 8.94368 — 



TO FIND THE MOON^S APPAR. LONG. AND LAT. 301 



tf 


# 


v 5° 1' 10" . 


cos. 9.99833 






44 54.4. 2694 /, .4 . 


«. 3.43213 


V 


log. 3.43046 


h 


.. 


-o • •* 


tan, 9.87725 


X' . 


• 


• . . -. *, - 


tan. 8.94368- 


K + fr 




. 47 37 22 . . 

—2 0.2 * 120".2 „ 


cos. 9.82867 


z 


log. 2.08006 


v—z . 


. 


46 54.6 




v—z( sign changed) —46 54.6 




True lat. 


; 


. —4 14 14.7 




Appar. lat. 


5 1 9.3 S.' 




,P 






log. 3.18050 


X 


- 


.. ar. co* cos. 0.00119 


u 


+ 


. ar. cc 


!. log. 6.82242 


X' 


<c 


....*. 


cos. 9.99833 


& . 


' 


15 24,2 . 924 /; .2 . 


log. 2.96577 



Augm. semi-diana. 15 29.4 . 929".4 . log, 2.96821 

Exam. 2. About the middle of the eclipse of the sun on the 18th 
of September, 1838, the moon's longitude was 175° 29' 19".0, 
latitude 47' 47".5, equatorial parallax 53' 53".5, and semi-diame- 
ter 14' 41". 1 ; and the longitude of the nonagesimal at New York 
was 216° 20' 50" 5 the altitude 32° 15' 48": required the apparent 
longitude and latitude, and the augmented semi-diameter of the 
moon. 

Equat. paral. 53' 53".5 Moon's long. 175° 29' 19 v 

Reduction, 4 .4 Long, nonag. 216 20 50 



Sun's 


paral. 
P = 

■* . 

a • 


53 49 .1 
8 .6 


K 

h-- 

x = 

ar. co 
5 . 

• 


= -40 51 31 
= 32 15 48 


• 
P 
h 
X 


=■53 40 .5 

. 3220".5 . 

32° 15' 48" . 

47 47.5 . 

45 23.5 . 2723". 
32 15 48 . 

-40 51 31 . 

—18 45 . 1125" 


= O 47 47.5 

log. 3.50792 

cos. 9.92716 

. cos. 0.00004 


X 

h 


log. 3.43512 
tan. 9.80023 


K 


c. 3.23535 
sin. 9.81570— 


u 


log. 3.05105— 



302 



K-fw. 


• 


-41° 10' 16" 
-18 52.9 


.Li ri\\JDL»\ 

. 1132' 


CUYlCr. 

r .9 . 


c. 3.23535 
sin. 9.81844— 


v! 


log. 3.05379- 


K+u f . 


> 


-41 10 24 

-18 52.9 
175 29 19.0 


. 1132' 
144".0 


.9 . 

ar. 
ar. 


c. 3.23535 
sin. 9.81844— 


P 

True long. 


log. 3.05379— 


Appar. long 

P 

X 

u 

X— x . 


175 10 26.1 
2'24".0 . 


. log. 3.05379 
co. cos. 0.00004 
co. log. 6.94895 

. log. 2.15836 


Appar. latitude 


2' 24".9 N. 


144".9 


• 


. log. 2.16114 


P 

X 

u 
R 


• 


14'41".l . 


881".l 


ar. 
ar. 


. log. 3.05379 
co. cos. 0.00004 
co. log. 6.94895 

. log. 2.94502 


Augm. semi- 


■diam. 14 46 .7 . 


886".7 


. 


. log. 2.94780 



PROBLEM XVIII. 

To find the Mean Right Ascension and Declination, or Longitude 
and Latitude of a Star •, for a given time, from the Tables, 

Take the difference between the given year and 1840. Then 
seek in Table XY for the fraction of the year answering to the 
given month and days, and add it to this difference, if the given, 
time is after the beginning of the year 1840 ;. but if it is before, 
subtract it. Multiply the sum or difference by the annual variation 
given in the catalogue, (Table XC, or XCII,) and. the product will 
be the valuation in the interval between the given time and the 
epoch of the catalogue. Apply this product to the quantity given 
in the catalogue, according to its sign, if the given time is after 
the beginning of the year 1 840, but with the opposite sign if it is 
before, and the result will be the quantity sought. 

Exam. 1. Required the mean right ascension and declination of 
the star Sirius on the 1 5th of August, 1 842. 

Interval between given time and beginn. of 1840, (t y ) 2.619yrs. 
Annual variation of right ascension, . . . 2.646s» 

Variation of right ascension for interval t y . . 6.93s. 



TO FIND A STAR'S ABERR. IN RIGHT ASCENSION, ETC. 303 

A similar operation gives for the variation of declination in the 
same interval, 11 ".65. 

Mean right ascen., beginning of 1840, Table XC, 6 h - 38 m - 5n&- 
Variation for interval t, . + 6.93 



Mean right ascension required, . . 6 38 12.69 

Mean declination, beginning of 1840, . . 16° 30' 4".79S. 

Variation for interval t, . . . . . + 1 1 .65 



Mean declination required, . . „ . 16 30 16 .44 S. 

2. Required the mean longitude and latitude of Aldebaran on 
the 20th of October, 1838. 

Interval between given time and begin, of 1840, (t) 1.200yrs. 

Annual variation of longitude, .... 50".210 

Variation of longitude for interval t, 60". 2 

A similar operation gives for the variation of latitude in the same 
interval 0".4. 

Mean longitude, beginning of 1840, . 2 s - 7° 33' 5".9 

Variation for interval t, , . . — 10.2 



Mean longitude required, . . . 2 7 32 5 .7 

Mean latitude, beginning of 1840, . . 5° 28' 38" .0 S. 

Variation for interval t, . . . . + .4 



Mean latitude required, . . . . , 5 28 38 A S." 

3. Required the mean right ascension and declination of Capella 
on the 9th of February, 1839 ? 

Ans. Mean right ascension 5 k 4 m - 48.74% and mean declination 
45° 49' 38".53 N. 

4. Required the mean longitude and latitude of Aldebaran on 
the 16th of April, 1845? 

Ans. Mean longitude 2 s - 7° 37' 31".4, and mean latitude 5° 28' 
36".2. 



PROBLEM XIX. 

To find the Aberrations of a Star in Right Ascension and Decli- 
nation, for a given Day. 

This problem may be resolved for any of the stars in the cata- 
logue of Table XC by means of the following formulae : 



304 ASTRONOMICAL PROBLEMS. 

log. (aber. in right ascen.) .= M + log. sin ( O + <j>) — 10. 
log. (aber. in declin.) == N -f- log. sin (O + d) — 10, 

in which M, N, are constant logarithms, O the longitude of the sun 
on the given day, and <p, 6, auxiliary angles. M, N, and the an- 
gles 9, 3, are given for each of the stars in the catalogue, in 
Table XCI. O may be derived from an ephemeris of the sun, 
or it may be computed from the solar tables by Problem IX. 

Exam. 1. What was the amount of aberration, in right ascen- 
sion and declination, of a Orionis on the 20th of December, 1 837, 
the sun's longitude on that day being 8 s - 28° 28' ? 

Right Ascension. 
Table XCI, 9 . 6 s - 3° 13' M. . 0.1361 

O . 8 28 28 



O + 9 . 3 1 41 . .sin. 9.9998 



Aberration = 1".37 . . . .log. 0.1359 

Declination. 
Table XCI, . 8 s - 28° 23' N . . 0.7521 

O . 8 28 28 



O + d . 5 26 51 . . sin. 8.7399 



Aberration = 0".31 . . . .log. 1.4920 

2. Required the aberrations in right ascension and declination 
of a Andromedae on the 1st of May, 1838, the sun's longitude be- 
ing l 8 10° 38'. 

Ans. Aberr. in right ascension — 1".07, and aberr. in declina- 



tion - 11 ".69. 



PROBLEM XX. 

To find the Nutations of a Star in Right Ascension and Declma- 
tion y for a given Day. 

This Problem may be solved by means of the formulae, 
log. (nuta. in right asc.) = M' + log. sin (Q + ?') — 10 ; 
log. (nuta. in declin.) = N' -f log. sin {Q + &') • — 10 ; 

in which M', N ; , are constant logarithms, Q the mean longitude of 
the moon's ascending node, and <p', 6', auxiliary angles. M', N', 
and the angles <p', 6', are given for each of the stars in the cata- 
logue, in Table XCI. The mean longitude of the moon's ascend- 
ing node is given for every tenth day of the year in the Nautical 
Almanac, page 266, and may be easily found for any intermediate 



TO FIND A STAR'S NUTATION IN RIGHT ASCEN., ETC. 305 

day from the daily motion inserted at the foot of the column of 
longitudes. It may also be had by finding the supplement of the 
moon's node, for the given time, from the lunar tables, and sub- 
tracting it from 12 s - o° r. 

Exam. 1. What was the amount of the nutation, in right ascen- 
sion and declination, of a Orionis on the 20th of December, 1837, 
the mean longitude of the moon's node on that day being 1 8° 54' ? 



Right Ascension. 
Table XCI, <p' . 6 9 - 0° 15' M' . 
Q . 18 54 


. 0.0481 


Q, + ?'. 6 19 9 . 


sin. 9.5159 — 


Nutation = - 0".37 

Declination. 
Table XCI, 6' . 3 s - 2° 37' N' . 
fl . 18 54 


log.T5640- 
. 0.9657 


fl+tf'. 3 21 31 . 


sin. 9.9686 


Nutation = 8".60 


log. 0.9343 



2. Required the nutations in right ascension and declination of 
a Andromedae on the 1st of May, 1838. 

Ans. Nutation in right ascension — 0".54, and nutation in de- 
clination - 1".43. 

Note. When the apparent place of a star is desired with great 
accuracy, the solar nutations must also be estimated and allowed 
for. These may be determined by repeating the process for find- 
ing the lunar nutations, only using twice the sun's longitude in 
place of the longitude of the moon's node, and multiplying the re- 
sults by the decimal .075. 

The calculation of the solar nutations m Example 1st, is as fol- 
lows : 



Right Ascension. 
Table XCI, <p' . . 6 s - 0° 15' M' . 
20 . . 5 26 56 


. 0.0481 


2 + <p' . 11 27 11 . 


sin. 8.6914— 


-0".05 . 
.075 


log.~2.7395- 


Solar Nutat. = - 0".00 
39 





306 ASTRONOMICAL PROBLEMS. 



Table XCI,d' . 
20 


Declination. 

3 s - 2° 37' N' . 

5 26 56 


. 0.9657 


2O + 0' . 


8 29 33 . 

— 9".24 . 
.075 


sin. 10.0000— 




0.9657— 



Solar Nutat. = - 0".69 

In Example 2d, we find for the solar nutation in right ascension, 
— 0".08, and for the solar nutation in declination, — 0".51. 



PROBLEM XXI. 

To find the Apparent Right Ascension and Declination of a Star, 
on a given Day. 

Find the mean right ascension and declination for the given day 
by Problem XVIII ; then compute the aberrations in right ascen- 
sion and declination by Problem XIX, and the lunar and solar nu- 
tations m right ascension and declination by Problem XX. Apply 
the aberrations and nutations according to their signs, to the mean 
right ascension and declination on the given day, observing that the 
declination when south is to be marked negative, and the results 
will be the apparent right ascension and declination sought. 

Exam. 1. What was the apparent right ascension and declina- 
tion of a Orionis on the 20th of December, 1837 ? 

h. m. s. ° ' " 

Table XC, M. right ascen. 5 46 30.71 M. dec. 7 22 17.14N. 
Variations . -6.59 . . —2.42 





5 46 24.12 


Aberr. . 


+ 1.37 


Lun. nutat. . 


— 0.37 


Sol. nutat. 


0.00 



7 22 14.72 
+ 0.31 
+ 8.60 
-0.69 



App. right asc. 5 46 25.12 App.dec. 7 22 22.94N. 

2. Required the apparent right ascension and declination of 
a Andromedae on the 1st of May, 1838. 

Ans. Appar. right ascen. Oh. Om. 0.90s., and appar. dec. 
28° 11' 39".92. 



TO FIND A STAR'S ABERRATION IN .LONGITUDE, ETC. 307 



PROBLEM XXII. 

Tb find the Aberrations of a Star in Longitude and Latitude, for 
a given Day. 

The formulae for the computation are, 

log. (aber. in long.) = 1.30880 + log. cos (6s. + O — L) + ar. 

co. log. cos X — 10; 
log, (aber. in lat.) = 1.30880 + log. sin (6s. + O - L) -f log. 

sin X — 20 ; 

in which O = longitude of the sun on the given day ; L = mean 
longitude of the star ; and X = mean latitude of the star. 

Exam. 1. Required the aberrations in longitude and latitude of 
Antares on the 26th of February, 1838, the sun's longitude on thai 
day being 11 s - 7° 29'. 

By Prob. XVIII, L = 8 s - 7° 30', and X = 4° 32' S. 
6s.iO . 17 7 29 Const log. 1.3088 

6s. + O - L 8 29 59 . . cos. 6.4637 — 
X . . 4 32 . ar. co. cos. 0.0014 



Aberr. in long. = — 0".00 . log. 3.7739 — 

Const, log. 1.3088 
6s. + O - L 8 s - 29° 59 7 . . sin. 10.0000 - 
X . . . 4 32 . . sin. 8.8978 



Aberr, in lat. = - 1".61 . log, 0.2066 — 
2, Required the aberrations in longitude and latitude of Arc- 
fairus on the 5th of October, 1838, the sun's longitude being 
6 B - 11° 47', 

Ans, Aberr. in long. — 23". 34, and aberr. in lat. 1".85. 
Note. The nutation in longitude of a fixed star may be found 
after the same manner as the nutation in longitude of the sun. 
See Problem IX,) 

PROBLEM XXIII, 

To find the Apparent Longitude and Latitude of a Star, for a 

given Day. 

Find the mean longitude and latitude on the given day by Prob» 
iem XVIII. Find also the aberrations in longitude and latitude by 
Problem XXII, and the nutation in longitude, as in Problem IX. 
Apply the aberration and nutation in longitude, according to their 



308 ASTRONOMICAL PROBLEMS". 

signs, to the mean longitude, and the result will be the apparent 
longitude ; and apply the aberration in latitude according to its 
sign, to the mean latitude, and the result will be the apparent 
latitude. 

Exam, 1 . Required the apparent longitude and latitude of An- 
tares on the 26th of February, 1838. 

Table XC, M. long. 8 s " 7° 31' 45".2 M.. lat.. 4° 32' 51",6 S. 
Var. . . — 1 32 .57 . . .78 





8 


7 


30 


12 


.63 


, . 4 


32 50 


.82 


Aberr. 


, 









.00 


r .- 


-1 


.61 


Nutat. 


- 






-4 


.40 









App.long.8 7 30 8 .23 App. lat. 4 32 49 .21 & 

2. Required the apparent longitude and latitude of Arcturus on 
the 5th of October, 1838. 

Ans, Appar.long. 6 8 - 21° 58' 37".4, and appar. lat.30° 51' 19".K 

PROBLEM XXIV. 

To compute the Longitude and Latitude of a Heavenly Body from 
its Right Ascension and Declination, the Obliquity of the Eclip- 
tic being given. 

This Problem may be solved by means of the following for- 
mulae : 

log. tang x = log. tang D + ar. co. log. sin R ; 

log. tang L=log. cos (x— w) + log. tang R + ar. co. log. cos a?— 10; 

log. tang X = log. tang (x — w) + log. sin L — 10 y 

in which 

R = the Right Ascension ; 

D = the Declination (minus when South) * r 

L = the Longitude ; 

X — the Latitude ; 

w = the Obliquity of the ecliptic - r 

x is an auxiliary arc. It must be taken according to the sign of 
its tangent, but always less than 1 80°. The longitude will always 
be in the same quadrant as the right ascension. The latitude must 
be taken less than 90°, and will be north or south, according as the 
sign is positive or negative. 

Note. When the mean longitude and latitude are to be derived 
from the mean right ascension and declination, the mean obliquity 
of the ecliptic is taken. When the apparent longitude and latitude 
are to be derived from the apparent right ascension and declina- 
tion, found as in Problem XXI, the apparent obliquity is taken. 



TO COMPUTE THE RIGHT ASCEN. AND DEC, OF A BODY. 309 

The mean obliquity of the ecliptic at any assumed time is easily 
deduced from Table XXII. The apparent obliquity is found by 
Problem X. 

Exam. 1. On the 20th of June, 1838, the right ascension of 
Capella was 76° 11' 29", the declination 45° 49' 35" N., and the 
obliquity of the ecliptic 23° 27' 37" ; required the longitude and 
latitude. 

D = 45° 49' 35" . . . tan, 0.0125295 
R = 76 11 29 . . ar. co. sin. 0.0127367 



x = 46 39 56 . . „ tan. 0.0252662 
w = 23 27 37 



& _ M = 23 12 19 . . . cos. 9.9633623 
R= 76 11 29 . . . tan. 0.6094483 
x = 46 39 56 . . ar. co. cos. 0.1635240 



Long. = 79 36 4 . . . tan. 0.7363346 

L = 79 36 4 . • , sin. 9.9928075 
x — w = 23 12 19 . , . tan. 9.6321632 



Lat. = 22 51 49 . . . tan. 9.6249707 

2. Given the right ascension of Spica 199° 11' 35", and decli- 
nation 10° 19' 24" S., and the obliquity of the ecliptic 23° 27' 36", 
on the 1st of January, 1840, to find the longitude and latitude, 
Ans, Long. 201° 36' 32", and lat. 2° 2' 30" S. 



PROBLEM XXV. 

To compute the Right Ascension and Declination of a Heavenly 
Body from its Longitude and Latitude, the Obliquity of the 
Ecliptic being given. 

The formulae for the solution of this problem are, 

log. tang y = log. tang X -f- ar. co. log. sin L ; 

log. tang R =log. cos(y + w) + log. tang L + ar. co. log. cos y— 10; 

log. tang D = log. tang (y + w) + log. sin R — 10 ; 

in which 

L = the Longitude ; 

X = the Latitude (minus when South) ; 

R = the Right Ascension ; 

D = the Declination ; 

w = the Obliquity of the ecliptic ; 

y is an auxiliary arc. It must be taken according to the sign of 
its tangent, but always less than 180°. The right ascension will 



310 ASTRONOMICAL PROBLEMS. 

always be in the same quadrant with the longitude. The declina- 
tion must be taken less than 90°, and will be north or south, ac- 
cording as the sign is positive or negative. 

Note. The mean or apparent obliquity of the ecliptic is taken, 
according as the given and required elements are mean or apparent. 

Exam. 1. On the 1st of January, 1830, the longitude of Sirius 
was 3 s - 11° 44' 18", the latitude 39° 34' 1" S., and the obliquity 
of the ecliptic 23° 27' 41" : required the right ascension and de- 
clination. 

X = — 39° 34' 1" . . tan. 9.9171381 - 
L = 101 44 18 . ar. co. sin. 0.009178S 



y = 139 50 14 . . tan. 9.9263169— 
u = 23 27 41 ■ 



^-{-w = 163 17 55 . . , . cos. 9.9812819— 
L = 10144 18 . . tan. 0.6823798— 
y = 139 50 14 . ar. co. cos. 0.1 167843 — 



Right ascen = 99 24 48 . „ tan. 0.7804460— 

R= 99 24 48 . . sin. 9.9941121 
y + co = 163 17 55 . . tan. 9.4771803— 



Dec. = 16 29 20 S. . . tan. 9.471 2924— 

2. Given the longitude of Aldebaran 67° 33' 5", and latitude 
5° 28' 38" S., and the obliquity of the ecliptic 23° 27' 36", on the 
1st of January, 1840, to find the right ascension and declination. 

Ans. Right ascension 66° 41' 4", and declination 16° 10'57"N. 



PROBLEM XXVI. 

The Longitude and Declination of a Body being given, and alsa 
the Obliquity of the Ecliptic, to find the Angle of Position. 

The formula is 

log. sin p = log. sin w + log. cos L + ar. co. log. cos D — 10 : 

p = Angle of Position (required) ; 

L = Longitude ; 

D = Declination ; 

w == Obliquity of the ecliptic. 

The angle of position p must be taken less than 90°. It Is to be 
observed also that when the longitude is less than 90°, or more 
than 270°, the northern part of the circle of latitude lies to the west 
of the circle of declination, but that when the longitude is between 
90° and 270°, it lies to the east. 

Note. The angle of position may also be computed from the 



TO FIND THE TIME OF NEW OR FULL MOON. 311 

right ascension and latitude, by means of a formula similar to that 
just given, namely, 

log. s'mp = log. sin w + log. cos R + ar. co. log. cos X — 10; 
Exam. 1. Given the longitude of Regulus 147° 27' 54", and 
declination 12° 47' 45" N., and the obliquity of the ecliptic 23° 
27' 41", to find the angle of position. 

u = 23° 27' 41" . . sin. 9.6000260 
L = 147 27 54 . . cos. 9.9258601 
D= 12 47 45 . ar. co. cos.0.0109217 



Angle of pos. = 20 7 58 . sin. 9.5368078 

The circle of latitude lies to the east of the circle of declination. 
2. Given the longitude of Fomalhaut 331° 27' 56", and declina- 
tion 30° 31' 14" S., and the obliquity of the ecliptic 23° 27' 41", 
to find the angle of position. Ans. 23° 57' 20". 

The circle of latitude lies to the west of the circle of declination. 



PROBLEM XXVII. 

To find from the Tables the Time of New or Full Moon, for a 
given Year and Month. 

For New Moon. 

Take from Table LXXXVI, the time of mean new moon in 
January, and the Arguments I, II, III, and IV, for the given year. 
Take from Table LXXXVII, as many lunations with the corre- 
sponding variations of Arguments I, II, III, and IV, as the given 
month is months past January, and add these quantities to the for- 
mer, rejecting the ten thousands from the sums in the columns of 
the first two arguments, and the hundreds from the sums in the 
columns of the other two. Seek the number of days from the first 
of January to the first of the given month, in the second or third 
column of Table LXXXVIII, according as the given year is a 
common or bissextile year, and subtract it from the sum in the col- 
umn of mean new moon : the remainder will be tabular time of 
mean new moon for the given month. It will sometimes happen 
that the number of days taken from Table LXXXVIII, will ex- 
ceed the number of days of the sum in the column of mean new 
moon : in this case one lunation more, with the corresponding ar- 
guments, must be added. 

With the sums in the columns I, II, III, and IV, as arguments,, 
take the corresponding equations from Table LXXXIX, and add 
them to the time of mean new moon : the sum will be the Approxi- 
mate time of new moon for the given month, expressed in mean 
time at Greenwich. 

Next, for the approximate time of new moon calculate the true 
longitudes and hourly motions in longitude of the sun and moon; 



312 



ASTRONOMICAL PROBLEMS. 



subtract the less longitude from the greater, and the hourly mo- 
tion of the sun from the hourly motion of the moon ; and say, as 
the difference between the hourly motions : the difference between 
the longitudes : : 60 minutes : the correction of the approximate 
time. The correction added to the approximate time, when the 
sun's longitude is greater than the moon's, but subtracted, when 
it is less, will give the true time of new moon required, in mean 
time at Greenwich. This time may be reduced to the meridian 
of any given place by Problem V. 

For Full Moon. 

Take from Table LXXXVI, the time of mean new moon, and 
the corresponding Arguments I, II, III, and IV, for January of the 
given year, and from Table LXXXVII, a half lunation with the 
corresponding changes of the arguments. Then, when the time 
of mean new moon for January is on or after the 16th, subtract the 
latter quantities from the former, increasing, when necessary to 
render the subtraction possible, either or both of the first two argu- 
ments by 10,000, and of the last two by 100 ; but add them when 
the time is before the 16th. The result will be the tabular time 
of mean full moon and the corresponding arguments, for January. 
Proceed to find the approximate time of full moon after the same 
manner as directed for the new moon. 

For the approximate time of full moon calculate the true longi- 
tudes and hourly motions in longitude of the sun and moon. Sub- 
tract the sun's longitude from the moon's, adding 360° to the latter 
if necessary. Take the difference between the remainder and VI 
signs, and call the result R. Also subtract the hourly motion of 
the sun from the hourly motion of the moon. Then say, as the 
difference between the hourly motions : R : : 60m. : the correction 
of the approximate time. The correction added to the approxi- 
mate time of full moon, when the excess of the moon's longitude 
over the sun's is less than VI signs, but subtracted when it is- 
greater, will give the true time of full moon. 

Exam. 1. Required the time of new moon in September, 1838^ 
expressed in mean time at New York. 



1838, 
81un. 


M. New Moon. 


I. 


n. 


in. 


IV. 


d. 

24 

236 


h- m. 

16 53 
5 52 


0681 
6468 


9175 
5737 


99 
22 


85 
93 


Days, 


260 
243 


22 45 


7149 


4912 


21 


78 


Sept'r, 

I. 

II. 

III. 

IV. 


17 


22 45 

16 

9 35 

3 

10 




Sept'r. 


18 


8 49 


Appro. 


cimate t 


ime, 



TO FIND THE TIME OF NEW OR FULL MOON. 



313 



Moon's true long, found for approx. time, is 5 s - 25° 29' 19" 
Sun's do. do. do. 5 25 27 27 



Difference, 1 52 

Moon's hourly motion in long, is 29 28 

Sun's do. do. ... 2 27 

Difference, ...... 27 1 

As 27' 1" : 1' 52" : : 60 m - : 4 m - 9 s -, the correction. 

Approx. time of new moon, September, . 18 d - 8 h< 49 m - B 
Correction, ...... — 4 9 

True time, in mean time at Greenwich, . 18 8 44 51 
Diff. of meridians, . . . . . 4 56 4 

True time, in mean time at New York, . 18 3 48 47 

Exam. 2. Required the time of full moon in April, 1838, ex- 
pressed in mean time at New York. 



1838, 
$ lun. 


M. Full Moon. 


I. 


II. 


III. 


IV. 


d. h. m. 

24 16 53 

14 18 22 


0681 
404 


9175 
5359 


99 

58 


85 
50 


3 lun. 


9 22 31 

88 14 12 


0277 
2425 


3816 
2151 


41 
46 


35 

97 


Days, 


98 12 43 

90 


2702 


5967 


87 


32 


April, 

I. 

II. 

III. 

IV. 


8 12 43 

8 29 

16 7 

15 

30 




April, 


9 14 4 


Approj 


dmate t 


irae. 





Moon's true long, found for approx. time, is 6 s * 19° 44' 17" 



Sun's 



do 



do. 



do. 



Moon's hourly motion in long, is 
Sun's do. do, 

Difference 

As 27' 48" : 1' 5" : : 60 m - : 2 m - 20 s -, the correction. 

40 



19 


45 


22 


5 29 

6 


58 



55 



R. . 


1 


5 




30 
2 


15 
27 



27 48 



314 ASTRONOMICAL PROBLEMS. 

Approximate time of full moon, April, . 9 d * 14 h * 4 m - 8 - 
Correction, + 2 20 



True time, in mean time at Greenwich, . 9 14 6 20 
Diff. of meridians, 4 56 4 



True time, in mean time at New York, . 9 9 10 16 

3. Required the time of new moon in September, 1837, ex- 
pressed in mean time at Philadelphia ; taking the longitudes for 
the approximate time from the Nautical Almanac. 

Ans. 29d. 3h. Om. 5s. 

4. Required the time of full moon, in October, 1837, expressed 
in mean time at Boston. Ans. 13d. 6h. 30hi. 25s. 



PROBLEM XXVIII. 

To determine the number of Eclipses of the Sun and Moon that 
may be expected to occur in any given Year, and the Times 
nearly at which they will take place. 

For the Eclipses of the Sun. 

Take, for the given year, from Table LXXXVI the time of 
mean new moon in January, the arguments and the number N. 
If the number N differs less than 37 from either 0, 500, or 1000, 
an eclipse must occur at that new moon. If the difference is be- 
tween 37 and 53, there may be an eclipse, but it is doubtful, and 
the doubt can only be removed by a calculation of the true places 
of the moon and sun. If the difference exceeds 53, an eclipse is 
impossible. 

If an eclipse may or must occur at the new moon in January, 
calculate the approximate time of new moon by Problem XXVII, 
and it will be the time nearly of the middle of the eclipse, express- 
ed in mean time at Greenwich. This may be reduced to the 
meridian of any other place by Problem V. 

To find the first new moon after January, at which an eclipse 
of the sun may be expected, seek in column N of Table LXXXVII 
the first number after that answering to the half lunation, that, 
added to the number N for the given year, will make the sum come 
within 53 of 0, 500, or 1000. Take the corresponding lunations, 
changes of the arguments, and the number N, and add them, re- 
spectively, to the mean new moon in January, the arguments, and 
the number N, for the given year. Take from the second or third 
column of Table LXXXVIII, according as the given year is a 
common or bissextile year, the number of days next less than the 
days of the sum in the column of mean new moon, and subtract it 
from this sum ; the remainder will be the tabular time of mean 
new moon in the month corresponding to the days taken from Ta- 



TO FIND THE NUMBER OF ECLIPSES IN A YEAR. 



315 



ble LXXXVIII. At this new moon there may be an eclipse of 
the sun ; and if the sum in the column N is within 37 of the num- 
bers mentioned above, there must be one. Find the approximate 
time of new moon, and it will be the time nearly of the middle of 
the eclipse. 

If any of the other numbers in the last column of Table 
LXXXVII are found, when added to the number N of the given 
year, to give a sum that falls within the limit 53, proceed in a simi- 
lar manner to find the approximate times of the eclipses. 

Note. When the sum of the numbers N, or the number N itself, 
in case the eclipse happens in January, is a little above 0, or a 
little less than 500, the moon will be to the north of the sun, and 
there is*a probability that the eclipse will be visible at any given 
place in north latitude at which the approximate time of the eclipse, 
found as just explained and reduced to the meridian of the place, 
comes during the day-time. When the number N found for the 
eclipse is more than 500, the moon will be to the south of the sun, 
and the eclipse will seldom be visible in the northern hemisphere, 
except near the equator. 

For the Eclipses of the Moon. 

Find the time of full moon and the corresponding arguments and 
number N, for January of the given year, as explained in Problem 
XXVII. Then proceed to find the times at which eclipses of the 
moon may or must occur, after the same manner as for eclipses of 
the sun, only making use of the limits 35 and 25, instead of 53 
and 37.* 

Note. An eclipse of the moon will be visible at a given place, 
if the time of the eclipse thus found nearly, and reduced to the 
meridian of the place, comes in the night. 

Exam. 1. Required the eclipses that maybe expected in the 
year 1840, and the times nearly at which they will take place. 

For the Eclipses of the Sun. 





M. New Moon. 


I. 


II. 


III. 


IV. 


N. 


d. 


h. 


m. 












1840, 


3 


10 


30 


0085 


6386 


65 


63 


844 


2 1un. 


59 


1 


28 


1617 


1434 


31 


98 


170 




62 


11 


58 


1702 


7820 


96 


61 


014 




60 






As the sum of the numbers N 


March, 


2 


11 


58 


I. 




8 


3 


comes within 37 of 0, there must be 


II. 




19 


38 


an eclipse. 


III. 






12 




IV. 






13 




March, 


3 


16 


4 


Mean 


time at 


Green\ 


rich. 





* The numbers 53, 37, and 35, 25, are the lunar and solar ecliptic limits, as 
determined by Delambre. The limits given in the text, converted into thousandth 
parts of the circle, are 55, 37, and 37, 21. 



316 



ASTRONOMICAL PROBLEMS. 





M. 2s ew Moon. 


i. n. 


in. 


rv. 


N. 


d. 


h. 


m. 










1840, 


3 


10 


30 


00S5 , 6386 


65 


63 


844 


8 Ian. 


236 


5 


52 


6468 5737 22 


93 

1 


682 




239 


16 


22 


6553 1 2123 | 87 


56 : 


526 




213 






As the sum of the numbers N 


August, 


26 


16 


22 


I. 







54 


comes within 37 of 500, there 


must 


II. 







49 


be an eclipse. 




III. 






15 






IV. 






16 




J 


August, 


26 


18 


36 


Mean time at Greenwich 





For the Eclipses of the Moon 






1840, 
ilun. 


M. Full Moon. 


I- 


n. 


ni. 


IV. 


N. 


d. 

3 

14 


h. 
10 
18 


m. 

30 
22 


0085 
404 


6386 
5359 


65 
58 


63 
50 


844 
43 


llun. 


18 
29 


4 
12 


52 
44 


489 
808 


1745 
717 


23 

15 


13 

99 


887 
85 




47 
31 


17 


36 


1297 


2462 


38 ! 12 


972 

X, al- 
f 1000, 

eclipse 


Febr. 

I. 

II. 

III. 

IV. 


16 


17 
7 



36 

27 

23 

5 

27 


As the sum of the numbers 
though it comes within 35 o 
does not come within 25. the 
may be considered doubtful. 


Febr. 


17 


1 


58 


Mean 


time at 


Greenv 


rich. 





M. F 


all Moon. 


i. n. | in. 


IV. 


N. 


d. 


h, 


m. 










1840. 


18 


4 


52 


489 ! 1745 


23 


13 


887 


7 lun. 


206 


17 


8 


5659 5020 


7 


94 


596 




224 


22 





6148 I 6765 | 30 


07 


433 




213 






As the sum of the numbers N 


August, 


11 


22 





I. 




1 


37 


comes within 25 of 500, there must 


II. 




19 


16 


be an eclipse. 


III. 






3 




IV. 






25 




August, 


12 


19 


21 


Mean time at 


Greenw 


ich. 





2. Required the eclipses that may be expected in the year 1839, 
and the times nearly at which they will take place, expressed in 
mean civil time at New York. 



TO CALCULATE A LUNAR ECLIPSE. 



317 



Axis. One of the sun on the 15th of March, at 9h. 20m. A. M. ; 
and one of the sun on the 7th of September, at 5h. 24m. P. M. 

3. Required the eclipses that may be expected in the year 1841, 
and the times nearly at which they will take place, expressed in 
mean civil time at New York. 

Ans. Four of the sun, namely, one on the 22d of January, at 
12h. 18m. P. M. ; one on the 21st of February, at 6h. 17m. A.M. ; 
one on the 18th of July, at 9h. 24m. A. M. ; and one on the 16th 
of August, at 4h. 28m. P.M.: and two of the moon, namely, one 
on the 5th of February, at 9h. 10m. P. M>; and one on the 2d of 
August, at 5h. 5m. A. M. 

The eclipses of the sun in January and August may be con- 
sidered as doubtful. 



PROBLEM XXIX. 

To calculate an Eclipse of the Moon, 

The calculation of the circumstances of a lunar eclipse is effect- 
ed with the following fundamental data, derived from the tables of 
the sun and moon : 



Approximate Time of Full Moon (at Greenwich) 


, T 


Sun's Longitude at that time, . 


L 


Do. Hourly Motion, 


, 




s 


Do. Semi-diameter, 


t , 




5 


Do. Parallax, 


, 




P 


Moon's Longitude, . 


. 




, I 


Do. Latitude, 


, 




X 


Do. Equatorial Parallax, 


, 




P 


Do. Semi-diameter, 


. , 




d 


Do. Hourly Motion in longitude, . 




771 


Do. Hourly Motion in latituc 


le, . . 




n 



We obtain the time T by Problem XXVII ; the quantities ap- 
pertaining to the sun, namely, L, s, and 6, by Problem IX ;* and 
those which have relation to the moon, namely, I, X, P, d, m, and 
7i, by Problem XIV. 

From these quantities we derive the following : 

True Time of Full Moon, (at given place,) 
Moon's Latitude at that time, .... 
Semi-diameter of earth's shadow, 
Inclination of Moon's relative orbit, . 

T being known, T' is found as explained in Problem XXVU. 
To obtain X', we state the following proportion, 

1 hour : correction for the time of full moon : : n : x ; 
* p may be taken =• 9". 



T' 

V 
S 
I 



318 ASTRONOMICAL PROBLEMS. 

from this we deduce the value of x ; and thence find X by the 
equation 

\> =\±x. 
When the true time of full moon, expressed in mean time at 
Greenwich, is later than the approximate time, the upper sign is 
to be used, if the latitude is increasing, the lower if it is decreas- 
ing ; but when the true time is earlier than the approximate time, 
the lower sign is to be used if the latitude is increasing ; the upper 
if it is decreasing. 

The value of S is derived from the equation 

S=(P+p-6) + «\(P+p-5); 
and the angle I from the formula 

log. tang I = log. n +ar. co. log. (m — s). 
The foregoing quantities having all been determined, the various 
circumstances of the eclipse may be calculated by the following 
formulae : 

For the Time of the Middle of the Eclipse. 

3.55630 + log. cos I + ar. co. log. (m — s) — 20 = R ; 
log. t = R + log. X' + log. sin I — 10 ; 
M = T' ± t : 
t = interval between time of middle of eclipse and time of full 
moon ; M = time of middle of the eclipse. 

The upper sign is to be taken in the last equation when the lati- 
tude is decreasing ; the lower, when it is increasing. 

For the Times of Beginning and End. 
log. c = log X' + log. cos I — 10 ; 
l 0g . u _ log-(S + ^+c) + log.(S+^-c) ( R . 

B = M — v, and E = M + v : 
v = half duration of the eclipse ; B = time of beginning ; and E = 
time of end. 

Note. If c is equal to or greater than S + d, there cannot be an 
eclipse. 

For the Times of Beginning and End of the Total Eclipse. 
l 0g . v> = log.(S-d + c) + log.(S-d--c) + R . 

B' = M — *;', and E' = M + v' : 
v' = half duration of the total eclipse ; B' = time of beginning of 
total eclipse ; and E' = time of end of total eclipse. 

Note. When c is greater than S — d, the eclipse cannot be total. 

For the Quantity of the Eclipse. 
log. Q = 0.77815 + log. (S + d - c) + ar. co. log. J — 10; 
Q = the quantity of the eclipse in digits. 



TO CALCULATE A LUNAR ECLIPSE. 



319 



Note 1 . An eclipse of the moon begins on the eastern limb, and 
ends on the western. In partial eclipses the southern part of the 
moon is eclipsed when the latitude is north, and the northern part 
when the latitude is south. 

Note 2. When the eclipse commences before sunset, and ends 
after sunset, the moon will rise more or less eclipsed. To obtain 
the quantity of the eclipse at the time of the moon's rising, find 
the moon's hourly motion on the relative orbit by the equation 

log. h = log. (m — s) -f ar. co. log. cos I ; 

in which h = hourly motion on relative orbit. Also find the inter- 
val between the time of sunset and the time of the middle of the 
eclipse, which call i. Then, 

1 hour : i : : h : x. 
Deduce the value of x from this proportion, and substitute it in 
the equation 

in which c designates the same quantity as in previous formulae. 
Find the value of c', and use it in place of c in the above formula 
*br the quantity of the eclipse, and it will give the quantity of the 
eclipse at the time of the moon's rising. When the eclipse begins 
before and ends after' sunrise, the quantity of the eclipse at the 
time of the moon's setting may be found in the same manner, only 
using sunrise instead of sunset. 

Example. Required to calculate, for the meridian of New York, 
the eclipse of the moon in October, 1 837. 



Elements. 




Approximate time of full moon, 


T = 


ll h - 10 m -(Oct. 13) 


Sun's longitude at that time, . 


L = 


6 s - 20° 24' 28" 


Do. hourly motion, 




s = 


2 29 


Do. semi-diameter, 




6 = 


16 4 


Do. parallax, 




P = 


9 


Moon's longitude, . 




I = 


20 21 51 


Do. latitude, 




X = 


11 28 S. 


Do. equatorial parallax, 




P = 


59 32 


Do. semi-diameter, 




d = 


16 13 


Do. hourly motion in long. 




m = 


35 54 


Do. hourly motion in lat. (tending nc 


>rth), n = 


3 19 


Approx. time of full moon, October, 




13 d - ll h - 10 m - 00 8 - 


Correction found by Prob. XX 


VII, 


. 


+ 4 42 



True time, in mean time at Greenwich, 
Diff. of meridians, .... 



13 11 14 42 
4 56 4 



True time, in mean time at New York, T' = 13 6 18 38 



320 



ASTRONOMICAL PROBLEMS. 



60 m. . 4 m. 42 s. . . 3' 


19" : x = 


16" 


. 


Moon's lat. at approx. time, 
Correction. .... 


• . 




X = 11' 28" S. 
x= —16 


Moon's lat. at true time, 


X'=ll 12 


Moon's equatorial parallax, . 
Sun's do 


• 




. P=59'32" 
. p = 9 


Sum, 

Sun's semi-diameter, . 


eV(P+P 


59 41 

. 8 =16 4 


Diff. 

Add 


y — 5 =43 37 

— 6) = 44 


Semi-diameter of earth's shadow, 


. 




. S = 44 21 


Moon's hor. mot. less sun's (m — s) 
Moon's hor. motion in latitude, n 


= 2005" . 
= 199 . 


ar. 


co. log. 6.69789 
. log. 2.29885 


Inclination of rel. orbit, 1 = 5° 40' 


# . 




. tan. 8.99674 


Time of I\ 

I 5° 

m — s .... 


liddle. 

40' . 
2005" 

672" 
40' . 

;■• = us 9 - 
IP.M. 


ar. 


3.55630 

cos. 9.99787 

co. log. 6.69789 


V 

I 5° 


R. 0.25206 
. sin. 8.99450 


t ()h. im. 55 

T' . 6 18 35 


. log. 2.07393 



Middle, 



S+d + c 
S + d— c 



6 20 36 P.M. 

Times of Beginning and End. 



11' 9" = 669" 



log. 2.82737 
cos. 9.99787 



4303" 
2965 



!*• 46 a - 22* f « 6382 fc 



jog. 2.82524 

log. 3.63377 
log. 3.47202 

2 ) 7.10579 

3.55289 
R. 0.25206 



log. 3.80495 





TO CALCULATE A SOLAR ECLIPSE 


321 


17 

Middle, . 


l h - 46 m - 22 s - = 6382 s - 
6 20 36 


log. 3.80495 


Beginning, . 
End, 


4 34 14 P. M. 
8 6 58 P. M. 




S-d+c 
S-d-c 


. 2357" 
. 1019 


log. 3.37236 
log. 3.00817 

2 ) 6.38053 




h 46 m - 9 s - = 2769 s - 
6 20 36 


3.19026 
R 0.25206 


v' 
Middle, 


log. 3.44232 



Beg. of total eclipse, 5 34 27 P. M. 
End of total eclipse, 7 6 45 P. M 



S+d-c 
d 

Quantity, 



973" 



0.77815 

log. 3.47202 

ar. co. log. 7.01189 



18.3 digits, 



log. 1.26206 



PROBLEM XXX. 

To calculate an Eclipse of the Sun, for a given Place. 

Having found by the rule given in the note to Problem XXVIII, 
that there is a probability that the eclipse will be visible at the 
given place, and calculated the approximate time of new moon by 
Problem XXVII, find from the tables, for this time or for the near- 
est whole or half hour, the sun's longitude, hourly motion, and 
semi-diameter ; and the moon's longitude, latitude, equatorial par- 
allax, semi-diameter, and hourly motions in longitude and latitude. 
Find also by Problem XVI, the longitude and altitude of the 
nonagesimal degree ; anS thence compute by Problem XVII, the 
apparent longitude, latitude, and augmented semi-diameter of the 
moon, (using the relative horizontal parallax.) With these data 
compute the apparent distance of the centres of the sun and moon, 
at the time in question, by means of the following formulae : 

log. tang 6 = log. X' + ar. co. log. a ; 
log. A = log. a + ar. co. log. cos 6 ; 
41 



322 ASTRONOMICAL PROBLEMS. 

in which 

A = appar. distance of centres ; 
X' = app^r. Lat. of Moon ; 

a = Diff. of appar. Long, of Moon and Sun = diff. of appai 
long, of Moon (found as above) and true long, of Sun. 

6 is an auxiliary arc. The value of 6 being derived from the 
first equation, the second will then make known the value of A. 

a and X' are in every instance to be affected with the positive 
sign.* 

For the Approximate Times of Beginning, Greatest Obscuration, 

and End. 

Let the time for which the above calculations are made, be de- 
noted by T. If the apparent distance of the centres of the sun 
and moon, found for the time T, is less than the sum of their ap- 
parent semi-diameters, there is an eclipse at this time. But if it 
is greater, either the eclipse has not yet commenced, or it has al- 
ready terminated. It has not commenced if the apparent longitude 
of the moon is less than the longitude of the sun ; and has termi- 
nated, if the apparent longitude of the moon is greater than the 
longitude of the sun. 

1. If there should be an eclipse at the time T, from the sun's 
longitude and hourly motion in longitude, and the moon's longi- 
tude and latitude, and hourly motions in longitude and latitude, 
found for this time, calculate the longitudes and the moon's lati- 
tude for two instants respectively an hour before, and an hour after 
the time T. The semi-diameter of the sun, and the equatorial 
parallax and semi-diameter of the moon, may, in our present in- 
quiry, be regarded as remaining the same during the eclipse. Find 
the apparent longitude and latitude, and the augmented semi-diam- 
eter of the moon, (using in all cases the relative parallax,) and 
thence compute by the formulae already given, the apparent dis 
tance of the centres of the sun and moon at the two instants in 
question. 

Observe for each result, whether it is less or greater than the 
sum of the apparent semi-diameters of the two bodies. If the 
moon is apparently on the same side of the sun at the times T and 
T -f lh., take the difference of the distances of the two bodies in 
apparent longitude at these times, but, if it is on opposite sides, 
take their sum, and it will be the variatj^n of this distance in the 

* A, the apparent distance of the centres, may be found without the aid of loga- 
rithms by means of the following equation : 

A _ V a? -f A'2. 

If the logarithmic formulae are used, it will be sufficient here to take out the angle 
6 to the nearest minute. When we have occasion to obtain the distance of the 
centres exact to within a small fraction of a second, must be taken to the nearest 
4ens of seconds, if it exceeds 20° or 30°. 



TO CALCULATE A SOLAR ECLIPSE. 323 

hour following T. Find in like manner the variation of the dis- 
tance during the hour preceding T. Then, if the apparent distance 
of the centres at the times (T — lh.), (T -f- Ik) is less than the 
sum of the apparent semi-diameters, deduce from these results 
the variations of the distance in apparent longitude during the pre- 
ceding and following hours, allowing for the second difference, and 
observing whether the two bodies are approaching each other, or 
receding from each other. Thence, find the distance in apparent 
longitude at the times (T — 2h.), (T + 2h.) Find by the same 
method the apparent latitude of the moon at the instants (T — 2h.), 
(T + 2h.), observing that the variation of the apparent latitude in 
any given interval is the difference between the latitudes at the 
beginning and end of it, if they are both of the same name ; their 
sum, if they are of opposite names. 

From these results derive the apparent distance of the centres 
of the sun and moon at the two instants in question. 

If there should still be an eclipse at the time (T + 2h.) or 
(T — 2h.), find by the same method the distance of the centres at 
the time (T + 3h.) or (T — 3h.) These calculations being effect- 
ed, the times of the beginning, greatest obscuration, and end of the 
eclipse, will fall between some of the instants T,(T— lh.),(T -f- lh.)» 
&c, for which the apparent distance of the centres is computed. 

2. If the eclipse occurs after the time T, the different phases 
will happen between the instants T, (T + lh.), (T + 2h.), &c. 
Find the apparent distance of the centres of the sun and moon for 
the times (T + lh.), (T + 2h.), by the same method as that by 
which it is found for the times (T + lh.), (T — lh.), in the case 
just considered. Then, if the eclipse has not terminated, deduce 
the distance of the moon from the sun in apparent longitude, and 
the moon's apparent latitude, for the time (T +3h.), from these 
distances and latitudes at the times T, (T -f- lh.), (T + 2h.) ; as 
in the preceding case the distance and latitude for the time 
{T-f2h.) were deduced from the same at the times (T — lh.), T, 
(T + lh,) With the results obtained compute the apparent dis- 
tance of the centres of the two bodies at the time (T + 3h.) 

3. In case the eclipse occurs before the time T, jhe apparent 
distance of the centres must be found bv similar methods for the 
times (T - lh.), (T - 2h.), &c. 

The calculation is to be continued until the distance, from being 
less, becomes greater than the sum of the semi-diameters. 

Now, let h = variation of apparent distance of centres in the 
interval of one hour comprised between the first two of the instants 
for which the distance is computed ; d = difference between the 
sum of the semi-diameters of the sun and moon and the apparent 
distance of their centres at the first instant ; and t = interval be- 
tween first instant and the time of the beginning of the eclipse. 
Then, 

hid:: 60 m - " t (nearly.) 



324 ASTRONOMICAL PROBLEMS. 

Find the value of t given by this proportion, and add it to the 
time at the first instant, and the result will be a first approximation 
to the time of the beginning of the eclipse, which call b. Find 7 
by interpolation,* the distance of the moon from the sun in appa- 
rent longitude (a), and the moon's apparent latitude (V), for this 
time, and thence compute the apparent distance of the centres. 
Take h = variation of apparent distance in the interval between the 
time b and the nearest of the two instants above mentioned, be- 
tween which the beginning falls, and d = difference between the 
apparent distance of the centres at the time b and the sum of the 
semi-diameters, and compute again the value of t. Add this to the 
time b, or subtract it from it, according as b is before or after the 
beginning, and the result will be a second approximation to the 
time of the beginning, which call B. A result still more approxi- 
mate may be had, by taking h — variation of apparent distance of 
centres in the interval B — 6, d = difference between apparent dis- 
tance at the time B and sum of semi-diameters, finding anew the 
value of t given by the preceding proportion, and adding it to, or 
subtracting it from, as the case may be, the time B. But prepara- 
tory to the calculation of the exact times, it will suffice, in general, 
to take the first approximation. 

The end of the eclipse will fall between the last two of the 
several instants for which the apparent distance of the centres of 
the moon and sun have been computed. The approximate time 
of the end is found by the same method as that of the beginning.! 

* The second differences may easily be taken into the account in finding the 
quantities a and V for the time b. Thus, let k = variation of a for the interval of 
an hour comprised between the instants above mentioned, A/ = same for the suc- 
ceeding hour, and i = interval between 6 and the nearer of the two instants, (in 

£ £ £/ 

minutes.; Then, if we put /= — , c = — — — , and v = var. of a in interval i, 

© do 



t 

v =— 



{/*(•+»! 



10 

The upper sign is to be used when the time b fs nearer the first than the second 
instant, the lower when it is nearer the second than the first, c is to be used with 
its sign. The error by this method will not exceed the number c, (supposing the 
changes of k, k 1 , from 10m. to 10m. to increase or decrease by equal degrees.) 

The general formula for interpolation is Q = q + - d' -} — ■= — d" 4- 

n~o~r^ d"' -\- &c, in which q is the first of a series of values, found at 

equal intervals, of the quantity whose value Q. at the time t is sought, t is reck- 
oned from the time for which q is found, h is one of the equal intervals, d', d", 
d"', &c, are the first, second, third, &c, differences. If we make h = 1, we have 

Q . , + * + «g=l> ,■ + ' (' ~l <' ~ 2) i- + & e. 

t In effecting the reductions of the quantities a and \' to the first approximate 
time of end, k 1 must stand for the variation of a during the hour preceding that 
comprised between the last two instants, and the last instant must be substituted 
for the first. (See Note above.) 



TO CALCULATE A SOLAR ECLIPSE. 



325 



The middle of the interval between the approximate times of 
the beginning and end of the eclipse, will be a first approximation 
to the time of greatest obscuration. 

Note. When the object is merely to prepare for an observation, 
results sufficiently near the truth may be obtained by a graphical 
construction. The elements of the construction are the difference 
of the apparent longitudes of the moon and sun, and the apparent 
latitude of the moon, found as above, for two or more instants du- 
ring the continuance of the eclipse. Draw a right line EF, (Fig. 
123,) to represent the ecliptic, assume on it some point C for the 

Fig. 123. 




position of the sun at the instant of apparent conjunction, and lay 
off CA, CA', equal to the two differences of apparent longitude ; 
and to the right or left, according as the moon is to the west or 
east of the sun at the instants for which the calculations have been 
made. Erect the perpendiculars Ap, A'p', and mark off A<z, A'a' 
equal to the two apparent latitudes. Through a, a', draw a right 
line, and it will be the apparent relative orbit of the moon, or 
will differ but little from it From C let fall the perpendicular Cm 
upon the relative orbit, in will be the apparent place of the moon 
at the instant of greatest obscuration. Take a distance in the di- 
viders equal to the sum of the apparent semi-diameters of the moon 
and sun, and placing one foot of it at C, mark off with the other 
the points /, /', for the beginning and end of the eclipse, and by 
means of a square mark on EF the points b, e, which answer to 
the beginning and end. If the eclipse be total or annular, mark 
the points of immersion and emersion, g, g', with an opening in 
the dividers equal to the difference of the semi-diameters, and find 
the corresponding points b', e' on the line EF. 

If the calculations are made from hour to hour, the distance AA' 
is the apparent relative hourly motion of the sun and moon in lon- 
gitude. This distance laid off repeatedly to the right and left will 
determine the points 1, 2, &c, answering to lh., 2h., &c. before 



326 ASTRONOMICAL PROBLEMS. 

and after the times for which the calculations are made. If the 
spaces in which the points b, e, answering to the beginning and 
end of the eclipse, occur, be divided into quarters, and then sub- 
divided into three equal parts or five-minute spaces, the approxi- 
mate times of the beginning and end of the eclipse will become 
known. 

From the point m, as a centre^ describe the lunar disc ; and 
from the point C, as a centre, describe the sun's disc, and we shall 
have the figure of the greatest eclipse. The quantity of the eclipse 
will result from the proportion 

SN : MN i : 12 : number of digits eclipsed. 

Draw from the centre C to the place of commencement/, the 
line Cf; and through the same point C raise a perpendicular to 
the ecliptic With the longitude of the sun at the time of the be- 
ginning, calculate its angle of position by Problem XIII, and lay it 
off in the figure, placing the circle of declination CP to the left if 
the tangent of the angle of position be positive, to the right if it be 
negative. 

Compute also for the time of beginning the angle of the vertical 
circle of the sun with the circle of declination, that is, the angle 
PSZ in Fig. 24, p. 47, for which we have in the triangle PSZ 
the side PS = co-declination, the side PZ = co-latitude, and the 
included angle ZPS. (The requisite formulae are given in the Ap- 
pendix.) Form this angle in the figure at the point C, placing CZ 
to the right or left of CP, according as the time is in the forenoon 
or afternoon ;. CZ will be the vertical, and Z the vertex, or highest 
point of the sun. The arc Zt on the limb of the sun will be the 
angular distance from the vertex of the point on the limb at which 
the eclipse commences. 

For the True Times of Beginning, Greatest Obscuration, and End. 
The approximate times of beginning, greatest obscuration, and 
end of the eclipse, being calculated by the rules which have been 
given, find from the tables, or from the Nautical Almanac, (see 
Problem XXXI,) the moon's longitude, latitude, equatorial paral- 
lax, semi-diameter, and hourly motions in longitude and latitude, for 
the approximate time of greatest obscuration.* With the moon's 
longitude and latitude, and hourly motions in longitude and latitude,, 
found for this time, calculate the longitude and latitude for the ap- 
proximate times of beginning and end. The parallax and semi- 
diameter may, without material error, be considered the same 
during the eclipse. With the moon's true longitude, latitude, and 
semi-diameter at the approximate times of beginning, greatest ob- 
scuration, and end, calculate its apparent longitude and latitude,, 

* It will, in general, suffice to calculate the moon's longitude and latitude front 
the elements already found for the approximate time of full moon, if these have 
been accurately determined. The equatorial parallax and semi-diameter may be 
found by interpolation from the Nautical Almanac. 



TO CALCULATE A SCLAR ECLIPSE. 327 

and augmented semi-diameter, for these several times, (making use 
of the relative parallax.) With the sun's longitude and hourly mo- 
tion previously found for the approximate time of new moon, find 
his longitude at the approximate times of beginning, greatest ob- 
scuration, and end. The sun's semi-diameter found for the ap- 
proximate time of new moon, will serve also for any time during 
the eclipse. With the data thus obtained, calculate by the formu- 
lae given on page 321 the apparent distance of the centres of the 
sun and moon at the approximate times of the three phases. 

Note. When very great accuracy is required, the moon's longi- 
tude, latitude, equatorial parallax, semi-diameter, and hourly mo- 
tions in longitude and latitude, must be calculated directly from 
the tables, or from the Nautical Almanac, for the approximate 
times of the beginning and end, as well as for that of the greatest 
obscuration. 

For the Beginning. 

Subtract the apparent longitude of the moon at the approximate 
time of beginning from the true longitude of the sun at the same 
time, and denote the difference by a. Do the same for the approx- 
imate time of greatest obscuration. Subtract the latter result from 
the former, paying attention to the signs, and call the remainder k. 
Next, take the difference between the apparent latitudes of the, 
moon at the approximate times of beginning and greatest obscura- 
tion, if they are of the same name ; their sum, if they are of oppo- 
site names ; and denote the difference or sum, as the case may be, 
by n. This done, compute the correction to be applied to the ap- 
proximate time of beginning by means of the following formulae : 
log. b = log. a -f- log. k -f- ar. co. log. n — 10 ; 

c =\> -b,S = d + 5-5"; 
log t = log. (S + A) -f log. (S — A) -f- ar. co. log. n + ar. 
co. log. c +log. L + 1.47712 - 20 : 
in which 

t = Correction of approx. time of beginn. (required) ; 

a = Diff. of appar. long, of Moon and Sun at approx. time; 

L= Half duration of eclipse in minutes (known approximately) ; 

k — Appar. relative motion of Sun and Moon in long, in the in- 
terval L ; 

n — Moon's appar. motion in lat. in same interval ; 

X'= Moon's appar. lat. ; 

d = Augmented semi-diameter of the Moon ; 

6 = Semi-diam. of Sun ; 

A = Appar. distance of centres of Sun and Moon. 

b and c are auxiliary quantities. 

First find the value of b by the first equation, and substitute it in 
the second. Then derive the values of c and S from the second 



328 ASTRONOMICAL PROBLEMS. 

and third equations, and substitute them in the fourth, and it will 
make known the value of t, which is to be applied to the approxi- 
mate time of the beginning of the eclipse according to its sign. 

The quantities a, k, n, &c, are all to be expressed in seconds. 
The apparent latitude X' must be affected with the negative sign 
when it is south. The motion in latitude, n, must also have the 
negative sign in case the moon is apparently receding from the 
north pole, a and k are always positive.* 

The result may be verified, and corrected, by computing the ap- 
parent distance of the centres at the time found, and comparing it 
with the sum of the semi-diameters minus 5". 

Note. When great precision is desired, the quantities k and n 
must be found for some shorter interval than the half duration of 
the eclipse. Let some instant be fixed upon, some five or ten 
minutes before or after the approximate time of the beginning of 
the eclipse, according as the contact takes place before or after. 
For this time deduce the longitude and latitude of the moon, from 
the longitude and latitude at the approximate time of beginning, 
by means of their hourly variations ; and thence calculate the ap- 
parent longitude and latitude, and the augmented semi-diameter. 
Find the longitude of the sun for the time in question, from its 
longitude and hourly motion already known for the approximate 
time of beginning. Then proceed according to the rule given 
above, only using the quantities thus found for the time assumed, 
in place of the corresponding quantities answering to the approxi- 
mate time of greatest obscuration. L will always represent the 
interval for which k and n are determined. 

For the End. 

Subtract the longitude of the sun at the approximate time of the 
end from the apparent longitude of the moon at the same time. 
Do the same for the approximate time of greatest obscuration. 
Then proceed according to the rule for the beginning, only substi- 
tuting everywhere the approximate time of the end for the approx- 
imate time of the beginning, and taking in place of the formula 
c—'k' — b, the following : 

c=\'+b. 

* It will be somewhat more accurate to use in place of k and n, as above de- 

fc £' £ £ fcl £ 

fined, the values of the following expressions : 2£ — — — or — — 3£ — — — , 

o oo b ou 

<n 7^' — — 71 71 Yl! 71 

— 2A — — — or — 3£ — — ; — . The first of each of these pairs of expressions 

o do o Jo 

is to be used in case the true time of beginning is after the approximate time; — 
the second in the other ease, k' and n' are the apparent relative motions in longi- 
tude and latitude during the last half of L. In case these expressions are used 
the following constant logarithm is to be employed instead of that above given, 
viz. 0.69897. 

In the calculation of the end of the eclipse, k and n will answer to the last half 
of L, and k' and n' to the first half. 



TO CALCULATE A SOLAR ECLIPSE. 329 



For the Greatest Obscuration. 

Take the sum of the distances of the moon from the sun in ap- 
parent longitude at the approximate times of the beginning and end 
of the eclipse, and call it k. Take the difference of the apparent 
latitudes of the moon at the same times,- if the two are of the same 
name ; but if they are of different names, take their sum. Denote 
the difference or sum by n. Let a' = the distance of the moon 
from the sun in apparent longitude at the true time of greatest ob- 
scuration ; X' = the apparent latitude of the moon at the approxi- 
mate time of greatest obscuration. 

k : n : : X' : a'. 

Find the value of a' by this proportion, affecting X', n, k, always 
with the positive sign. 

Ascertain whether the greatest obscuration has place before or 
after the apparent conjunction, by observing whether the apparent 
latitude of the moon is increasing or decreasing about this time ; 
the rule being, that when it is increasing, the greatest obscuration 
will occur before apparent conjunction ; when it is decreasing, 
after. If the approximate and true times of greatest obscuration 
are both before or both after apparent conjunction, from the value 
found for a' subtract the distance of the moon from the sun in ap- 
parent longitude at the approximate time ; but if one of the times 
is before and the other after apparent conjunction, take the sum of 
the same quantities. Denote the difference or sum by m. Also, 
let D = duration of eclipse, and t — correction to be applied to the 
approximate time of greatest obscuration. Then to find t, we have 
the proportion 

k : m : : D : t. 

If the apparent latitude of the moon is decreasing, t is to be 
applied according to the sign of m ; but if the apparent latitude is 
increasing, it is to be applied according to the opposite sign. 

A still more exact result may be had by repeating the foregoing 
calculations, making use now of the apparent latitude at the time 
just found. When the greatest accuracy is required, the values of 
k and n may be found more exactly after the same manner as for 
the beginning or end. 

For the Quantity of the Eclipse. 

Find by interpolation the apparent latitude of the moon at the 
true time of greatest obscuration. With this, and the distance in 
longitude 'a' obtained by the proportion above given, compute by 
the formulae on page 321, the apparent distance of the centres of 
the sun and moon at the time of greatest obscuration. Subtract 
this distance from the sum of the apparent semi-diameters of the 

42 



330 ' ASTRONOMICAL PROBLEMS. 

two bodies, diminished by 5", and denote the remainder by R 
Then, 

Sun's semi-diam. (diminished by 3") : R : : 6 digits : number of 
digits eclipsed. 

When the apparent distance of the centres of the sun and moon 
at the time of greatest obscuration is less than the difference be- 
tween the sun's semi-diameter and the augmented semi-diameter 
of the moon, the eclipse is either annular or total ; annular, when 
the sun's semi-diameter is the greater of the two ; total, when it 
is the less. 

For the Beginning and End of the Annular or Total Eclipse. 

The times of the beginning and end of the annular or total 
eclipse may be found as follows : the greatest .obscuration will take 
place very nearly at the middle of the eclipse in question, and will 
not differ, at most, more than five or eight minutes (according as 
the eclipse is total or annular) from the beginning and end : to 
obtain the half duration of the eclipse, and thence the times of the 
beginning and end, we have the formulae 

log. tang & =]og. V -far. co. log. a, log. k'=hg. k -f- ar. co. log. sin e ; 
S = 8-d- I", oiS=d — 6 + l"; 
loff c = log.(S+A) + log.(S-A) _ 
6- 2 ' 

log. t = ar. co. log. k' + log. c + log. D + 1.77815 — 10 ; 
Time of Begin. = M — t, Time of End = M + 1: 
in which 

M = Time of greatest obscuration ; 
X' == Moon's apparent latitude at that time ; 
a = Distance of moon from sun in appar. long. ; 
k = Variation of this distance during the whole eclipse, or rela- 
tive mot. in appar. long, during this interval ; 
k' = Moon's appar. mot. on relative orbit for same interval ; 
6 = Inclination of relative orbit ; 
<5 = Semi-diameter of sun ; 
d = Augm. semi-diam. of moon ; 
A = Appar. distance of centres ; 

D = Duration of eclipse, (partial and annular or total ;) 
t = Half duration of annular or total eclipse. 

The first value of S is used when the eclipse is annular, the 
second when it is total. The quantities may all be regarded as 
positive. The results may be verified and corrected by finding 
directly the apparent distance of the centres for the times obtained, 
and comparing it with the value of S. 



TO CALCULATE A SOLAR ECLIPSE. 331 

For the Point of the Surfs Limb at which the Eclipse commences. 

Find the angle of position of the sun, and the angle between its 
vertical circle and circle of declination, at the beginning of the 
eclipse, as explained at page 326. Let the former be denoted by 
p, and the latter by v. Give to each the negative sign, if laid off 
towards the right ; the positive sign if laid off towards the left. 
Let a - distance of the moon from the sun in apparent longitude 
at the beginning of the eclipse ; X' = the moon's apparent latitude 
at the same time ; and 6 — angular distance of the point of contact 
from the ecliptic. Compute the angle 6 by the formula 

log. tang 6 — log. X 7 + ar. co. log. a ; 

taking it always less than 90°, and positive or negative according 
to the sign of its tangent. X 7 is negative when south ; a is always 
positive. 

Let A == distance on the limb of the point of contact from the 
vertex. The above operations being performed, the value of A 
results from the equation 

A=p + v + 90° — 6; 

p, v, and fl being taken with their signs. 

If the result is affected with the positive sign, the point first 
touched will lie to the right of the vertex. If with the negative 
sign, it will lie to the left of the vertex. 

Note. The circumstances of an occultation of a fixed star by 
the moon may be calculated in nearly the same manner as those 
of a solar eclipse. The star in the occultation holds the place of 
the sun in the eclipse. The immersion and emersion of the star 
correspond to the beginning and end of the eclipse. The elements 
which ascertain the relative apparent place and motion of the moon 
and star, take the place of those which ascertain the relative appa- 
rent place and motion of the moon and sun. Thus the star's lon- 
gitude, corrected for aberration and nutation, (see Problem XXIII,) 
must be used instead of the sun's longitudes ; the apparent dis- 
tances of the moon from the star in latitude, instead of the moon's 
apparent latitudes ; and the moon's augmented semi-diameter, in- 
stead of the sum of the semi-diameters of the sun and moon. The 
difference of the longitudes, and the relative motion in longitude, 
must also now be reduced to a parallel to the ecliptic passing 
through the star, (see Art. 490, page 183.) If X = apparent lati- 
tude of star, a = diff. of appar. longitudes of moon and star, and 
k == relative motion in longitude, we must substitute in the formu- 
lae for the eclipse, for X',X 7 — X ; for a, a cos X ; and for k, k cos X. 
n will stand for the relative motion in latitude, or for the variation 
of X 7 — X. 

-Example. Required to calculate an eclipse of the sun, for the 



332 



ASTRONOMICAL PROBLEMS. 



latitude and meridian of New York, that will occur on the 18th of 
September, 1838. 

For the Approximate Times of the Phases. 
Approximate time of New Moon. 



Sept. 18 d - 8 h - 49 m - 




Sun's longitude, . 
Do. hourly motion, 
Do. semi-diameter, 


. 175' 


3 27' 31".4 

2 26 .7 

15 57 .0 


Moon's longitude, 
Do. latitude, 


. 175 


29 19 
47 47 


Do. equatorial parallax, 
Do. semi-diameter, 




53 53 
14 41 


Do. hor. mot. in long. 




29 29 


Do. hor. mot. in lat. . 




2 41 


Do. appar. long. (Prob. XVII), 

Do. appar. lat. (A'), 

Do. augm. semi-diameter, . 


. 175 


10 26 

2 25 N. 
14 47 


DifF. of appar. long, (a), 
Appar. dist. of cen. (A), 




17 5 
17 15 


Sum of semi-diameters, 




30 44 


7 h - 49 m> 






Sun's longitude, . 
Moon's appar. long. . 


. 175 
. 174 


3 25' 4" 
47 3 


Do. appar. lat. (V) 




8 12 N. 


Do. augm. semi-diameter, . 
DifF. of appar. long, (a), 
Appar. dist. of cen. (a), 




14 49 
38 1 
38 53 


Sum of semi-diameters, 




30 46 


gh. 49m. 






Sun's longitude, .... 


. 175< 


> 29' 58" 


Moon's appar. long. . 
Do. appar. lat. (V), 


. 175 


36 15 

2 18 S. 


Do. augm. semi-diameter, . 




14 44 


DifF. of appar. long, (a), 
Appar. dist. of cen. (a), 




6 17 
6 42 


Sum of semi-diameters, 




30 41 



7 h. 49 r 

8 49 

9 49 

10 49 



a 


diff. or h 


X' 


diff. or n. 


A 


diff. 


sura semi-d. 


2281" 

1025 

377 

1925 


1256" 

1402 

1548 


492" N 
145 N 
138 S 
357 S 


347" 

283 
219 


2333" 

1035 

402 

1958 


1298" 
1556 


1846" 
1844 
1841 
1839 



TO CALCULATE A SOLAR ECLIPSE. 

For the Approximate Time of Beginning. 
h = 1 298", d = 2333" - 1 846" = 487" ; 

1298" : 487" : : 60 m - : t = 22 m \5 

7 h - 49 m - 
22 



333 



1st Approxi. 8 h - IT 

7^49". . a = 2281" 
Corrections for 22 m - 447 



V=492"N. 

133 (See Note, p. 324) 



8 h. n m. 



a = 1834 



a = 1834" ar. co. log. 6.73660 . 
V = 359 . log. 2.55509 

& =11° 4' 30" . tan. ,9.29169 

Appar. dist. of cen. a = 1869" 
Sum of semi-diam. . 1846 



X' = 359 N. 

log. 3.26340 



ar. co. cos. 0.00817 



487" : 23' 
8 L 11' 
+ 1 



. log. 3.27157 
22 m - : t = l"- 2 s - 



2d Approxi. 8 h - 12" 1 * 
For the Approximate Time of the End, 

h =1556", d = 1958"- 1839" = 119". 

1556" : 119": : 60 m - : t = 4 m \6. 
10 h. 49m. 

-5 



1st Approxi. 10 h> 44 m - 

10*- 49 m - . a = 1925" 
Corrections for 5 m - 132 



X'=357"S. 
17 



10 h. 44m. 

a = 1793' 
X' = 340 



a = 1793 

ar. co. log. 6.74642 
. log. 2.53148 



V = 340 S. 
log. 3.25358 



. tan. 9.27790 . ar. co. cos. 0.00767 

Appar. dist. of cen. A = 1825" 
1839 



133" 



14' 



3.26125 
: 5 m - : t = 0°\5. 



ASTRONOMICAL PROBLEMS. 



10 b. 44m. 

.5 

2d Approxi. 10 h - 44 m \5 
For the Approximate Time of Greatest Obscuration, 



Approx. time of begin. 
Approx. time of end, 



. 8 h - 12 ra - 
. 10 44 

2 ) 18 56 



1st Approxi. . 9 28 
For the True Times of the Phases. 

Approx. time of Approx. time of Approx. time of 
Beginning. Greatest Obscur. End. 

8 h. 12 m. o> 28 m - 10 h - 44 m - 
Sun's longitude, 175° 26' 1".0 175° 29' 6".8 175°32' 12".6 
Do. semi-diam., 15 57 .0 15 57 .0 15 57 .0 
Moon's app. Ion. 174 55 36 .7 175 27 7 .7 176 2 17 .2 
Do. app. lat. 5 45 .3 N. 43 .5S. 5 32..4 S. 
Do.augm.semid. 14 48 .0 14 45 .1 14 41 .7 



8 h - 12 r 

9 28 

10 44 



a 


k 


V 


n 


A 


S 


1824 ,, .3 

119 .1 

1804 .6 


1705".2 
1923 .7 


345".3 N 
43 .5S 

332 .4S 


388".8 
288 .9 


1856".7 
1835 .0 


1840".0 
1833 .7 



For the True Time of Beginning. 





a 


. 


1824".3 


, 




h 


. 


1705 


.2 


. 




n 


• 


388 


.8 


• 




b = 


— 


8001 


.1 


. 






c = 


345 


.3 




X'- 


:8346 


.4 


. 


S-f 


A 


. 


3696 


.7 


, 


s- 


■ A 


. 


-16 


.7 


. 




n 


. 




. 


. 




L 


. 


. 


76m. 



. log. 3.26109 

. log. 3.23178 

ar. co. log. 7.41028- 



log. 3.90315- 



ar. co. log. 6.07850 
. log. 3.56781 
. log. 1.22272— 

ar.co.log. 7.41028— 
. log. 1.88081 

Const, log. 1.47712 



Con*, of approx. time, + 43 s - .4 



log. 1.63724 + 



TO CALCULATE A SOLAR ECLIPSE. 



335 



Corr. of approx. time, + 43 s - .4 

Approx. time, . 8 h - 12 m - .0 

True time of begin. 8 12 43 .4, in Greenwich time. 
Diff. of merid. . 4 56 4 



True time of begin. 3 16 39 .4, in New York time. 
For the True Time of End. 



a . . 1804". 6 
k . . 1923 .7 
n . . 288 .9 

b= -12016 .3 
V . - 332 .4 


10 h 


44 m 


. log. 3.25638 
. log. 3.28414 
ar. co. log. 7.53925— 

. log. 4.07977- 


\'+b=c= -12348 .7 
S +A . . 3668 .7 
S -A . . -1 .3 

n . . 

L . . . 76m. 

• 

Corr. of approx. time, 
Approx. time, 


ar. co. log. 5.90838— 
. log. 3.56451 
. log. 0.11394— 

ar. co. log. 7.53925- 

. log. 1.88081 
Const.log. 1.47712 

-3 s - . log. 0.48401 — 
.0 


True time of end, . 
Diff. of merid. 


10 
4 


43 
56 


57 .0, in Greenwich time. 
4 


True time of end, . 

For the True T 

True time of beginnir 
Do. of end, 


5 47 
one of Gi 

2d 

= 138" 
= 43 .5 


53, in New York time. 

"eatest Obscuration. 

. 8 h - 12 m - 43 s - .4 
. 10 43 57 .0 


9 h - 49" 1 - . . X' 

9 28 .V 


2) 18 56 40 .4 

Approx. 9 28 20 .2 

S. 
S. 



Diff. 21 



Diff. 94 .5 

21 m - : 20 s - : : 94".5 : 1".5 
43 .5 



9^ 28 m - 20 B - . 



45. 



ASTRONOMICAL PROBLEMS, 



1705".2 
1923 .7 


388".8 
288 .9 




ft =3628 .9 : 

Time of beginr 
Time of end. 


n = 677 .7 : 

t. S h - 12 m -43 s - 
10 43 57 


:X'=45 // ,0:a / = 8".4 

.4, at 9 h -25 nL a = 119".l 

.0 a' = 8 .4 


D = 


2 31 13 


.6 m = - 110 .7 



362S".9 : 110".7 : : 2 h - 31 m - 13 s - .6 : 4^ 36 8 - .8 

9 h 28 .0 



True time (nearly) 9 32 36 .8 

21 m. . 4 m. 37 s. . . 94 /- 5 . 2 q., i8 

43 .5 



At 9 h - 32 m - 37% V = 64 .3 

3628".9 : 677".7 : : 64".4 : 12".0 ; at 9 h " 32 m - 37% a = 8".4 

a' =12 .0 



771 = 3 .6 

362S".9 : 3".6 : : 2 L 31 m - 13 s \6 : 9 s - .0 
9 L 32 m - 36 .8 



9 32 27 .8 



True time of greatest obscur. 
DifT. of merid. 



9 h. 32 m. 27 s -. 8, in Greenw. time. 
4 56 4 



True time of greatest obscur. . 4 36 23 ,8, in N. Y. time. 
For the Quantity of the Eclipse. 

9h.32m.37s. v = 64 // .3 

21 m - :9 s -: : 94".5 : .6 



kt nearest approach of centres, 



V=63 .7 

a =12 .0 



a 



12".0 
63 .7 



ar.co.log. 8.92082, 
. 1.80414 



log. 1.07918 



tan. 0.72496, . ar. co. cos. 0.73253 



Shortest distance of centres, 64". 8 
Sum of semi-diameters, 1837 .0 



log. 1.81171 



1772 .2 

15 ; 54" : 1772". 2 : : 6 : 11.14 digits eclipsed. 



TO CALCULATE A SOLAR ECLIPSE. 337 

For the Situation of the Point at ivhich the Obscuration com- 
mences. 
8 h - 12 m - . . a =1824", . . X' = 345".3 N. 
76 m - : 43 s - : : 1705" : 16, 76 m - : 43 s - : : 389" : 3 .7 * 



Atthebeginn. . a = 1808, . . X' = 341 .6 

a . 1808 . ar. co. log. 6.74280 
X' . 341.6 . . log. 2.53352 

6 = 10° 41' 57" . . tan. 9.27632 

Obliq. eclip. (Prob.X), 23° 27' 47" . sin. 9.60005 . tan. 9.63753 

Sun's longitude, 175 26 3 . sin. 8.90093 . cos. 9.99862- 

sin. 8.50098, tan. 9.63615 - 
Sun's declination, 1° 49' 0" ; Angle of pos. 23° 23' 50". 
Meantime of begin. 3 h - 16 m - 39 s -, Lat. 40° 42' 40", Dec. 1°49' 0" 
Equa. of time, 5 58 90 90 

Appar, time, . 3 22 37, PZ =49 17 20,PS = 88 11 
60 



4 ) 202 37 



Hour angle P = 50° 39' 15" . cos. 9.80210 
Co. lat. PZ = 49 17 20 . tan. 0.06526 



77i = 36° 23' 0" . . tan. 9.86736 
Co. dec. PS =88 11 



m'= 51 48 
m=36 23 
P=50 39 15 



S =42 38 10 . . tan. 9.96413 

Angle of position, . ' . — 23° 23' 50" 

Angle from eclip. (6), . . — 10 41 50 
Angle of dec. circle from vertex (S), 42 38 10 

90 



ar. co. sin. 0.10466 
sin. 9.77320 
tan. 0.08627 



Angular dist. of point first touched from vertex, 98 32, to the right. 
For the Beginning and End of the Annular Eclipse. 

Approx. time, 9 h - 32 m - 27\8 =true time of greatest obscur. 
At this time, a = 12".2, X' = 63".7. 
a = 12".2 . ar. co. log. 8.91364 . . log. 1.08636 

X'=63 .7 . . log. 1.80414 

6 = 79° 9' 30" . tan. 0.71778 . ar. co. cos. 0.72564 



A = 64".9 . . log. 1.81200 
43 



338 , ASTRONOMICAL PROBLEMS. 

S + A^ISS"^ . log. 2.13290,0 =79° 9' 30" . ar. co. sin. 0.00783 
S - A = 6 .2 . log. 0.79239, /c=3628".9 . log. 3.55977 



2 ) 2.92529, k' . 

1.46264 
D=152 m - 


• 
.6 


ar. co. log. 6.43240 

1.46264 

.log. 2.18184 

Const, log. 1.77815 


* = h - l m - 11 s 


log. 1.85503 



Time of greatest obscur. . 4 36 23 .8 

Formation of ring, . . 4 35 12 .2, New York time. 

Rupture of do. . .4 37 35 .4 " " 



PROBLEM XXXI. 

To find the Mooris Longitude, Latitude, Hourly Motions, Equa- 
torial Parallax, and Semi-diameter, for a given time, from the 
Nautical Almanac. 

Reduce the given time to mean time at Greenwich ; then, 
For the Longitude. 

Take from the Nautical Almanac the calculated longitudes an- 
swering to the noon and midnight, or midnight and noon, next pre- 
ceding and next following the given time. Commencing with the 
longitude answering to the first noon or midnight, subtract each 
longitude from the next following one : the three remainders will 
be the first differences. Also subtract each first difference from 
the following for the second differences, which will have the plus 
or minus sign, according as the first differences increase or de- 
crease. 

Find the quantity to be added to the second longitude by rea- 
son of the first differences, by the proportion, 12 h - : excess of given 
time above time of second longitude : : second first difference : 
fourth term. 

With the given time from noon or midnight at the side, take from 
Table XCIII the quantities corresponding to the minutes, tens of 
seconds, and seconds, of the mean or half sum of the two second 
differences, at the top : the sum of these will be the correction for 
second differences, which must have the contrary sign to the mean 

The sum of the second longitude, the fourth term, and the cor- 
rection for second differences, will be the longitude required. 



TO FIND MOON'S LONG., ETC., FROM NAUTICAL ALMANAC. 339 



For the Latitude. 

Prefix to north latitudes the positive sign, but to south latitudes 
ihe negative sign, and proceed according to the rules for the lon- 
gitude, only that attention must now be paid to the signs of the first 
differences, which may either be plus or minus. 

The sign of the resulting latitude will ascertain whether it is 
north or south. 

For the Hourly Motion in Longitude. 

Solve the proportion, 12 h - : given time from noon or midnight 
: : half sum of second differences : a fourth term ; which must have 
the same sign as the half sum of the second differences. 

Take the sum of the second first difference, half the mean of 
the second differences, with its sign changed, and this fourth term, 
and divide it by 12 : the quotient will be the required hourly mo- 
tion in longitude. 

For the Hourly Motion in Latitude. 

With the given time from noon or midnight, the second first 
difference of latitude, and the mean of the second differences, find 
the hourly motion in latitude in the same manner as directed for 
finding the hourly motion in longitude. When the hourly motion 
is positive, the moon is tending north ; and when it is negative, 
she is tending south. 

For the Semi-diameter and Equatorial Parallax. 

The moon's semi-diameter and equatorial parallax may be taken 
from the Nautical Almanac, with sufficient accuracy, by simple 
proportion, the correction for second differences being too small to 
be taken into account, unless great precision is required. 

Corrections for Third and Fourth Differences. 

When the moon's longitude and latitude are required with great 
precision, corrections must also be applied for the third and fourtlf 
differences. To determine these, take from the Almanac the three 
longitudes or latitudes immediately preceding the given time, and 
the three immediately following it, and find the first, second, third, 
and fourth differences, subtracting always each number from the 
following one, and paying attention to the signs. With the given 
time from noon or midnight at the side, and the middle third 
difference at the top, take from Table XCIV the correction for 
third differences, which must have the same sign as the middle 
third difference when the given time from noon or midnight is less 
than 6 hours ; the contrary sign, when the given time is more than 
6 hours. 



840 ASTRONOMICAL PROBLEMS. 

With the given time, and half sum of fourth differences, take 
from Table XCV the correction for fourth differences, giving it 
always the same sign as the half sum. 

The sum of the third longitude or latitude, the proportional part 
of the middle first difference answering to the given time from 
noon or midnight, and the corrections for second, third, and fourth 
differences, having regard to the signs of all the quantities, will be 
the longitude or latitude required. 



APPENDIX. 



TRIGONOMETRICAL FORMULA 

I. Relative to a Single Arc or Angle a. 

1 . sin 8 a + cos 2 a = 1 

2. sin a = tan a cos <z 

tan a 

3. sin a = 

4- cos a — 

5. tan a = 

6. cot a = 

tan a sm a 

7. sin a = 2 sin J a cos ■§■ a 

8. cos # = 1 — 2 sin 2 \ a 

9. cos a = 2 cos 2 J a — 1 

sm a 
10. tan^a 



v'l + tan 2 a 


1 


^1 + tan 2 ^ 


sm a 


cos a 


1 _ cos a 



1L cot | a = 



1 + cos a 

sin a 
1 — cos a 



a . 1 — cos a 

12. tan 2 |« = -— 

1 + cos a 

13. sin 2 a = 2 sin a cos a 

14. cos 2 a = 2 cos 2 a — 1 = 1 — 2 sin 2 a 

II. Relative to Two Arcs a and 6, op which a is supposib 
to be the greater. 

15. sin (a + 6) = sin a cos 6 + sin b cos a 

16. sin (a — 6) = sin a cos b — sin 6 cos a 

17. cos (a + b) = cos a cos b — sin a sin b 

* The radius is supposed to be equal to unity in all of the formulae* 



342 APPENDIX, 

18. cos (a — ft) ■= cos a cos ft + sin a sin 6 

/ . 7 n tan a + tan 6 

19. tan (a + ft) = 



20. tan (a — b) 



29. 
30. 
31. 
32. 



38. 
39. 
40. 



1 — tan a tan h 
tan a — tan 6 



1 -Han a tan b 

21. sin a + sin 6 = 2 sin J (a + 6) cos | (a — h) 

22. sin a — sin b — 2 sin \ (a — fr) cos | (a + ft) 

23. cos a + cosft = 2 cos i (a + b) cos J (a — b) 

24. cos 6 — cos a = 2 sin J {a + b) sin \ (a — 6) 
sin (a + ft) 



25. tan a + tan 6 



cos a cos 6 



7 sin (a — 6) 

26. tan a — tan o = 7 

cos a cos b 

1 sin ( a + 6) 

27. cot a 4- cot b — -t— - — : — r' 

sin a sm 

7 sin (a — b) 

28. cot 6 — cot a = -t— - — ^r 

sin a sin 

sin a + sin b _ tan | (a + b) 

sin a — sm b tan | (a — b) 

cos 6 + cos a _ cot £ (a + 6) 

cos 6 — cos a tan | (a — b) 

tan a +tan b cot 6 + cot a _ sin (<z + b) 

tan a — tan b cot 6 — cot a sin (a — ft) 

cot ft — tan a _ cot a — tan b _ cos (a + b) 

cot 6 + tan a cot a + tan b cos (a — ft) 

33. sin 2 — sin 2 b = sin (a + b) sin (a — b) 

34. cos 2 a — sin 2 b = cos (a ■+■ b) cos (a — ft) 

35. 1 ± sin a = 2 sin 2 (45° ±|a) 

3 6 . L±!^=tan 2 (45°±ia) 
1 T sin a v 3 ' 

37. !± S -H^ = tan(45°±*a) 

cos a 



1 — sin a _ sin 2 (45° -|a) 
1 — cos a sin 2 £ a 

1 + sin ft = si n 2 (45° +jb) 
1 + cos a cos 2 \a 

1 + tan ft ,.- . , % 

— 1 : = tan (45° + ft) 

1 — tan ft v ' 



41. i^iH>-! = tan(45°-ft) 
1 + tan 6 



TRIGONOMETRICAL FORMULA. 



343 



42. sin a cos b = i sin (a -f- &) + £ sin (a — 5) 

43. cos a sin 6 = ^ sin (a + 6) — | sin (a — 6) 

44. sin a sin b —\ cos (a — b) — \ cos (a + b) 

45. cos a cos 6 = £ cos (a + 6) + J cos (a — b) 

III. Trigonometrical Series. 
a 3 a 



sin a 



cos a 



1 



46. <j 



tan a = a + 

1 



cot a = 



2.3 

a 2 
2 

3 

<2 

* 3 " 



+ 



2.3.4.5 

a 4 

2.3.4 

2a 5 



— &c. 



rf 



2.3.4.5.6 

17a 7 



+ &c. 



3^5 



3 2 . 5. 7 
2a 5 



3 3 . 5. 7 



+ &c. 
&c. 



Let a = length of an arc of a circle of which the radius is 1, and 
(a") = number of seconds in this arc, then to replace an arc ex- 
pressed by its length, by the number of seconds contained in it, we 
have the formula 

47. a = ( a ") sin 1" ; log. sin 1" =^6.685574867. 

IV. Differences of Trigonometrical Lines. 

48. A sin x = + 2 sin \ a x. cos (x -\- ± a x) 

49. A cos x = — 2 sin \ A x. sin (x + i A #) 
sin A a? 



50. a tan x = + 

51. Acot£ = — 



cos x* cos (x + A x) 
sin a a: 



sin x. sin (a: + A x) 
V. Resolution of Right-angled Spherical Triangles.* 
Table of Solutions. 

Given. Required. Solution. 

Hypothen. f side op. giv. ang. 52 sin x = sin h . sin a 

and <( side adj. giv. ang. 53 tan x = tan h . cos a 

an angle t the other angle 54 cot x =cosh . tan a 

rT the other side 55 cos £ = 

Hypothen. | cos 5 

and ^ ang. adj. giv. side 56 cos x = tan 5 . cot £ 

a side . ., __. sin 5 

ang. op. giv. side 57 sma? = - — - 

I sin ft 



* Daily's Astronomical Tables and Formulas. 



344 APPENDIX. 

sin s 



. , . the hypothen. 58 sin x = 
A side and sm « 

the angle ^ the other side 59 sin x = tan 5 . cot a 

opposite I , , . nn cos a 

the other angle 60 sm x = 

L 6 cos s J I 

A side and f the hypothen. 61 cot x = cos a . cot 5 

the angle ^ the other side 62 tanx = tan a . sin s 

adjacent [the other angle 63 cosx = sin a . cos s 

f the hypothen. 64 cos x = rectang. cos. of the 

The two <5 giv. sides 

sides [ an angle 65 cot x = sin adj. side x cot. 

op. side 

f the hypothen. 66 cos x = rectang. cot. of the 

The two J given angles 

"&* la side 67 cos x = C ° S ' ? p - ^ 

^ sm. adj.ang. 

In these formulae, x denotes the quantity sought. 
a = the given angle 
* = the given side 
h = the hypothenuse. 



The formulae for the resolution of right-angled spherical trian- 
gles are all embraced in two rules discovered by Lord Napier, and 
called Napier's Rules for the Circular Parts. The circular parts, 
so called, are the two legs of the triangle, or sides which form the 
right angle, the complement of the hypothenuse, and the comple- 
ments of the acute angles. The risht angle is omitted. In re- 
solving a right-angled spherical triangle, there are always three of 
the circular parts under consideration, namely, the two given parts 
and the required part. When the three parts in question are con- 
tiguous to each other, the middle one is called the middle part, and 
the others the adjacent parts. When two of them are contiguous, 
and the third is separated from these by a part on each side, the 
part thus separated is called the middle part, and the other two the 
opposite parts . The rules for the use of the circular parts are (the 
radius being taken = 1 ), 

1 . Sine of the middle part = the rectangle of the tangents of the 
adjacent parts. 

2. Sine of the middle part = the rectangle of the cosines of the 
opposite parts. 

PARTICULAR CASES OF RIGHT-ANGLED SPHERICAL TRIANGLES. 

Equations 52 to 67, or Napier's rules, are sufficient to resolve 
all the cases of right-angled spherical triangles ; but they lack pre- 
cision if the unknown quantity is very small and determined by 



RESOLUTION OF SPHERICAL TRIANGLES. 345 

means of its cosine or cotangent ; or, if the unknown quantity is 
near 90°, and given by a sine or a tangent : in these cases the fol- 
lowing formulae may be used : 
cos(B + C) 
68 ' ^^- cosiB-C) 

69. tan* 4 B =^^\ 

2 sin {a ■+- c) 

70. tan 2 ic = tani(a + &)tani(a — b) 

71 . tan (45° — 1&) = ^ tan (45° — x), tan x — sin a sin B 

72. tan 2 1&= tan (1^+45°) tan (?-jl?_45 ) # 

a is the hypothenuse, B, C, the acute angles, and b, c, the sides 
opposite the acute angles. 

VI. Resolution of Oblique-Angled Spherical Triangles. 
General Formula. 
Let A, B, C, denote the three angles of a spherical triangle, and 
a, b, c, the sides which are opposite to them respectively. 
„ n sin A sinB sin C 
sin a sin b sin c 
or, the sines of the angles are proportional to the sines of the op* 

posite sides. 

74. cos c = cos a cos b + sin a sin b cos C 

75. cos c = cos (a — b) — 2 sin a sin b sin 2 -^C 

76. cos C = sin A sin B cos c — cos A cos B 

77. sin a cos c = sin c cos a cos B + sin b cos C 

78. sin a cot c = cos a cos B -+■ sin B cot C 

79. sin a cos B = sin c cos b — sin b cos c cos A 

Case i. Given the three sides, a, b, c. 
To find one of the angles. 

80. sinnA = sin( *- 6 A )sin( *- C) 

sin o sm c 

or, 

81. cos 2 U = Sin * S ! n( *- a) 

sm o sm c 



Case ii. Given the three angles, A, B, C 

To find one of the sides. 

. . — cos K cos (K — A) 

83. sin 2 ia = ■ p • n 

2 sm Jd sin C 

44 



346 



APPENDIX. 



or, 



84 . co*fr=™< K - B > cw < K ^ 
sin B sin C 



85. K = 



A + B + C 



Case in. Given two sides a and b, and the included angle C. 
1°. To find the two other angles A and B. 

cos i(a — b)} 



86. tani(A + B)=cot|C. 



cos £ (a+b) I 



o-v i / i t*\ , ^ sin i ( a — &) I 

87. tan |(A - B) =cot ^C. . ' ; , .! \ 

2 J 2 smi {a+b)) 

2°. To find the third side c. 



Napier's Analogies. 



♦ i ♦ i/ M sin*(A + B) 



88. < 



or, 



tan lc= tan £(« + &). 



(A-B) 
cosi(A+B) 



cos \ (A-B) 
or equa. 73. 

Case iv. Given two angles A and B, and the adjacent side c. 
1°. To find the other two sides, a and b. 

89. ta n i(g + 6)=tanic. C0S * ( ^Tg ) ] 

cosi(A+B) . 

^ Napier s Analogies. 

90. tani( a -&)=tank. Si "!StlS 

sin i (A+ B) , 

2°. To find the third angle C. 



cotiC=tani(A-B). sin H a + *> 
sin i (a— b) 



91. ^ 



or, 



cot *C = tan } (A+ B) . ^ -£ 

cos | (a— 6) 

or equa. 73. 

Case v. Given two sides a, b, and an opposite angle A. 

To find the other opposite angle B ; take equation 73, or the 
proportion ; sines of the angles are as sines of the opposite sides. 
(For the methods of determining the remaining angle and side, see 
page 348, Case 3.) 

Case vi. Given two angles A, B, and an opposite side a. 
To find the other opposite side b ; sines of the angle are propor- 



RESOLUTION OF SPHERICAL TRIANGLES. 347 

tional to the sines of the opposite sides. (For the methods of de- 
termining the remaining side and angle, see page 348, Case 4.) 

OTHER METHODS OF RESOLVING OBLIQUE-ANGLED SPHERICAL 
TRIANGLES.* 

Except when three sides or three angles are given, the data 
always include an angle A, and the adjacent side b, besides a third 
part. The required parts in the different cases may be found by 
the following formulae, and formula 73. 



92. 


tan m — tan b cos 


A 




93. 


cot n = tan A cos b 


94. 


c=m + m' 






95. 


C=n + n' 


96. 


cos a cos m! 






97. 


cos A sin n 


cos b cos m 


cos B sinw' 


98. 


tan A sin m' 






99. 


tan a cos n 


tan B sin m 


tan b cos n' 




100. 


sin 


k 


= sin 


A sin b. 




From the angle C (Fig. 124) a perpendicular CD is let fall upon 
the opposite side c, which divides the 
triangle into two right-angled trian- 
gles, that are resolved separately. In 
the one, ACD, A and b are known, 
and it is easy to find the other parts, 
which, joined to the third given part, 
serve to resolve the second right-an- 
gled triangle BCD, and determine the /^~ --— ^ c p- 
unknown quantity required, m, m' A "~B 

denote the two segments of the base ; n, n' the two parts of the 
angle C ; and k the perpendicular arc CD. 

It must be observed, that if the perpendicular CD fell without 
the triangle, m and m', n and n' would have contrary signs ; this 
happens when the angles A and B at the base are of different kinds, 
(the one Z_,*the other >90°). When it is not known whether this 
circumstance has place or not, the problem is susceptible of two 
solutions. 

The detail of the different cases is as follows : the data are A, 
b, and another arc or angle. 

Case 1 . Given two sides and the included angle ; or b, c, A. 

Equation 92 makes known m, 94 m', which may be negative, 
(what the calculation shows,) 96 a, 98 B, and equation 73, (page 
345,) C, which is known in kind. 

Case 2. Given two angles and the adjacent side; or A, C, b. 

Equation 93 makes known n, 95 n' y which may be negative, 
(what the calculation shows,) 97 B, 99 a ; finally, equation 73 
(page 345) gives c, which is known in kind. 

* Francoeur's Practical Astronomy. 



348 APPENDIX. 

Case 3. Given two sides and an opposite angle; crb, a, A. 

Equation 92 gives m, 96 m' , 94 c, 98 and 73 B and C ; 

or else, 93 gives n, 99 n', 95 C, 97 and 73 B and c. 

This problem admits in general of two solutions. In effect, the 
arc m! or angle n' being given by its cos., may have either the 
sign + or — ; there are then two values for c, and also for C. w! 
and n' enter into equations 97 and 98 by their sines, whence result 
therefore also two values of B. 

Case 4. Given two angles, and an opposite side; or A, B, b. 

Equation 92 gives m, 98 m', 94 c, 96 a, and equation 73 makes 
known C ; 

or else 93 gives n, 97 n', 95 C, 99 and 73 a and c. 

There are also two solutions in this case ; for, m' or n' is given 
by a sin., and therefore two supplementary arcs satisfy the ques- 
tion. Thus c in 94, and a in 96, receive two values ; same for 
C in 95, and a in 99, &c. 

Instead of solving the two right-angled triangles, into which the 
oblique-angled triangle is divided, by equations 92 to 99, we may 
employ Napier's rules, from which these equations have been ob- 
tained. 

Isosceles Triangles. 

When the triangle is isosceles, B = C, b = c, the perpendicular 
arc must be let fall from the vertex A, and the equations furnished 
by Napier's rules, become very simple. We find 

101. sin i a = sin i A sin b 

102. tan £ a = tan b cos B 

103. cos b — cot B cot \ A 

104. cos ^ A = cos \ a sin B 

The knowledge of two of the four elements A, B, a, b, which 
form the isosceles triangle, is sufficient for the determination of the 
two others. 



INVESTIGATION OF ASTRONOMICAL FORMULA. 

Formula for the Parallax in Right Ascension and Declination, 
and in Longitude and Latitude. (See Article 120, page 55.) 

Let s (Fig. 125) be the true place 
of a star seen from the centre of the 
earth, s' the apparent place, seen from 
a point on the surface of which z is 
the zenith, the latitude being /. The 
displacement ss' = p is the parallax 
in altitude, which takes effect in the vertical circle zs' ; p is the 




PARALLAX IN RIGHT ASCENSION AND DECLINATION. 349 

pole ; the hour angle zps = q is changed into zps', and sps' = a 
is the variation of the hour angle, or the parallax in right ascen- 
sion ; the polar distance ps = d is changed into ps' ; the differ- 
ence S of these arcs is the parallax in declination or of polar dis- 
tance* We have, (For. 73, p. 345,) 

sin s' : s'm ps (d) : : sin sps' (a) : sin ss' (p), 

sin zps' (q -+-«) : sin zs' (Z) : : sin s' : sin pz (90°— I). 

Multiplying, term by term, we obtain 

sin s' sin (q -f- a ) • sin d sin Z : : sin a sin s' : sin p cos Z ; 

sin p cos Z . , , . 
whence, sin a = . , . — - sin (a + a) . 

sin a sin Z 

Or, substituting for p its value given by equa. (8,) p. 51, and 
replacing H by P, 

sin P cos I . . 

sin a = : — - — sin (<7 -r a ) . . . (A) . 

sin a 

This equation makes known a when the apparent hour angle 

zps' = q + a , seen from the earth's surface, is given ; but if we 

know the true hour angle zps = q, seen from the centre of the 

earth, developing sin (q + a), (For. 15, p. 341), and putting 

sin P cos I 
__ — = m 

sm a 

sin a = ?7i (sin ^ cos a + sin a cos §), 

or, dividing by sin a, 

1 = 77i (sin q cot a -|- cos <?) ; 

whence, by transformation, 

msinq „ . , . . 

tan a = — = m sin a -\- m' sin a cos a (very nearly.) 

1 — 771 COS q a Z 2\j j/ 

Restoring the value of m, 

sin P cos I . , /sinPcosZV . 
tan a = : — = — sm a + I i — - — I sm a cos q. 

sina 3 \ sin a / * * 

Putting the arc a in place of its tangent, and P in place of sin P, 
and expressing these arcs in seconds, (For. 47, p. 343,) there 
results, 

P cos I . /P cos l\z . ,,. _ x 

a = — : — y sm a + I — : — -j- I sm a cos q sm I" . . . (B). 
smd * \ smd / * * v ' 

The parallax in declination (5) is the difference of the arcs ps 
(= d) and jw' (=d + 6.) Let zs = z, and zs' = Z. The trian- 
gles zps and zps' give (For. 74 and 73), 

cos d — sin I cos z cos (d + 6) — sin / cos Z 

1°. cos ozs = J—. = i ' . , 

^ cos /sinz cos I sm Z 

• Francceur's Uranography, p. 418. 



, , , tx cos d smZ — sin 7 cos z sm Z , . , ._ 
cos (a -to) = : h sin / cos Z 

sm 2 



350 APPENDIX. 

sin flf sin q sin (<? + #) sin (q -f- a) 

2°. sm pzs = : i- = i — rr 

sm z sm Z 

From the first equation we derive 
cos d sin Z — sir 
sin z 
_ cos d sin Z — sin I (cos z sin Z — sin z cos Z) 

sinz 
_ cos d sin Z — sin I sin (Z — z) 
sin z 
or, (equ. 8, p. 51,) 

sinZ . . 

= — — (cos a — sin r sm /) ; 
sm z 

from the second, 

sinZ _ sin (d + ^) sin (q 4- a) 

sin z sin d sin # 

substituting, 

,,,,** sin (d + <5) sin (q + a ) , 7 • -^ • •* 

cos (<* + £)= — V-j-^ . y ' (cos rf-sinPsinA 

sm a sm q 

cos {d + 5) _ sin (^ + «) /cos d sin P sin Z\ 

sin (d + <5) sin q Vsin d sin d / 

, 7 , „ sin (a + a) / , sin P sin Z\ ,-_ 

cot (d + §) = V ' I cot d r— 5— ). . . (C). 

sin q \ sm d / 

^ sin P sin I 
rut tan x = : — =— ; 

sm a 

.i / 7 . t\ sin (q + a) , . 

then, cot (d -f (5) = ^ - (cot rf — tan x) 

sin </ 

_sin (g + a) /cosd sin #\ 
sin <? Vsin ^ cosx/ 

__ sin (<? + a) cos d cos a? — sin d sin # 
sin q sin </ cos # 

_ sin (q + a) cos (d + x) 



. . . (D). 



sin q sin d cos a: 

The apparent polar distance (d + 5) being computed by either 
of the formulae (C) and (D), we have S = (d + S) — d. 

Formulae may be obtained that will give the parallax in declina- 
tion without first finding the apparent declination, (except approx- 
imately.) 

From equa. (C) we obtain 

sin P sin I , sin q cot (d + 6) 
-—j— =cotd • / , ^ > 



PARALLAX IN RIGHT ASCENSION AND DECLINATION. 351 

and we also have 

i /j i i\ cose? cos (d + <5) sin 6 

COt a — COt (d+o)= — -j : — , , , n = - — j— : — , , , ». ,* 

v sin a sin (a + o) sin a sin (a + <)) 

the sum of these equations gives 

sin P sin Z , , , ~ /, sin c/ \ sin 5 

—3 — = cot ( d + 5 ) V l — -r—i — x\ i + • , • ( -, , xv 

sin a \ sin (c/ + a) / sin a sm (d + <)) 

sin a sin (a + a) — sin 7 

Now, 1 —. — ~~. = ^r-7 — r— r — - 

sm (q + a) sm (q + a) 

2 sin $ a cos (c/ + £ a) sin a cos (? + £ a) 

= r— 7 ■ r = -. j : (1 OX. 22, 1 3) 

sm (q + a) sm (q + a) cos £ a 

cos (q + £ a) sin P cos I . . . , 

= r— ^ j , by equa. (A;. 

sm a cos \ a ■* ^ 

Substituting, 

sin P sin / , , . „, cos (a + 2 a ) sin P cos Z , 

. , - = cot (d + S) —t— 7- — t ■ + 

sm a sm a cos i a 

sin 5 

sine? sin (d + ^)' 

or, sin <5 = sin P sin Z sin (d + 8) — 

cos (6? + 5) cos (q+^a) sinP cos Z .„. 

cos \<x 

= sin P sin Z [sin (d + 6) — tan y cos (d + 5)], 

cot Z cos (q -f- i a) 
making tany = ^^ ; 

whence, sin <5 = sin (d + 5 — y) . . . (F). 

cosy v *' v ' 

To facilitate the calculation, the sines of 8 and P in eqs. (E) 
and (F), may be replaced by the arcs. 

To obtain an expression for the parallax in declination in terms 

of the true declination, develope sin (d + 8 — y) in equation (F), 

which gives 

. x sin P sin Z . 

sm = [sm (d + <5) cos y— sm y cos (d + 0)] ; 

developing sin (6? + 8) and cos (d + 8), and reducing, we have 

sin P sin Z . 

sm = [sm (a — y) cos 5 -f- cos (a — y) sm 0] ; 

dividing by cos 8, 

. sin P sin Z ... , ... 
m = cosy [sm ( ^ + C0S ( d ~ y } laJi ^ 



352 APPENDIX. 

sin P sin 7 

cosy 
whence tan o — 



sin (d-y) 



sm P sm Z . , . 

1 cos (d—y) 

cosy v *' 



sin P sin Z . /7 v , /sin P sin A 2 

x 



. , , v . /sin P sinZY 

sin (d—y) + I ) 

V cos y / 



cos y \ cos y 

sin {d — y) cos (c?— y) (very nearly ;) 

or, replacing tan 5 and sin P by 5 and P, expressing these arcs in 

seconds, (For. 47, p. 343), and reducing by For. 13, p. 341, 

. PsinZ . ., . , /P sin Z\ 2 sin 1" . ,, . /in| . 

5 = sm (cZ— y) + 1 I — -z — sm2(d—y) . . . (G.) 

cosy x *' \cosy / 2 v ■ *' v ' 

If the place of a body be referred to the ecliptic, similar formu- 
lae will give the 'parallax in latitude and longitude, but as the 
ecliptic and its pole are continually in motion by virtue of the di- 
urnal rotation of the heavens, it is necessary, in order to be able to 
determine the parallax in longitude at any given instant, to know 
the situation of the ecliptic at the same instant. 

This is ascertained by finding the situation of the point of the 
ecliptic 90° distant from the points in which it cuts the horizon, 
and which are respectively just rising and setting, called the Non- 
agesimal Degree, or the Nonagesimal. 

Fig. 126. Let K (Fig. 126) be the 

pole of the ecliptic/6, p the 
pole of the equator fa ;/is 
the vernal equinox, the ori- 
gin of longitudes and of 
right ascensions ; libs is the 
eastern horizon, b the hor- 
oscope, or the point of the 
ecliptic which is just rising; 
pz = 90° — / (the latitude 
of given place) ; Kp =wthe obliquity of the ecliptic. The circle 
ULznv is at the same time perpendicular at n to the ecliptic fb, and 
at v to the horizon hb ; it is a circle of latitude and a vertical cir- 
cle, since it passes through the pole K and the zenith z : b is 90° 
from all the points of the circle Knv ; zn is the latitude of the ze- 
nith, /n its longitude ; the point n is the nonagesimal, since bn = 
90° ; nv is the altitude of this point, and the complement of zn ; 
nv measures the inclination of the ecliptic to the horizon at the 
given instant, or the angle b, so that b—nv — Kz ; thus fn = N 
the longitude of the nonagesimal, and nv =h the altitude of the 
nonagesimal, designate the situation of this point, and conse- 
quently ascertain the position of the ecliptic and its pole at the 
moment of observation.* 

* Francoeur's Uranography, p. 421. 




LONGITUDE AND ALTITUDE OF THE NONAGESIMAL. 353 

The points m and d are those of the equator and ecliptic which 
are on the meridian ; the arc fm, in time, is the sidereal time s, 
which is known ; the axcfi — 90°, since the plane Kpi, passing 
through the poles K and p, is at the same time perpendicular to 
the ecliptic and to the equator ; the arc mi =fi —fm = 90° — s ; 
then the angle zpK = 1 80° -zpi = 1 80° -mi = 90°+ s* 

Now, in the spherical triangle pKz we know the sides Kp =2 u, 
zp = 90°— I = H, and the included angle zpK — 90° -f- s ; and 
may therefore find Kz — h the altitude of the nonagesimal, and the 
angle pKz = nc —fc—fn — 90°— N = complement of the longi- 
tude N of the nonagesimal. Let S = sum of the angles Kzp and 
zKp, then, (For. 86, page 346,) 

tan |S - C0S ! / ( ^7i . cot i (90° + s), 
cos | (H+ w) 

tan|S = C ° B t5Sr"? - tan* (90"-*): 

cos i (H+ w) 

but, 

tan |S =— tan (180°— |S), and tan \ (90°— s) = — tan £ (5 — 90°) ; 
substituting, and denoting (180°— |S) by E, we have 

tanE = C ° S * ^j-") tan \ (5-90°) . . . (H). 
cos | (H+ w) 

Again, letD = zKp—Kzp, then, (For. 87,) 

,„ sin | (H — w) i / ^ . v 

tan |D = . \ )„ , \ . cot J (90° + s); 
sin | (H + w) v 

whence, by transforming as above, and denoting (180°— |D) by F, 
we have 

tanF = !Hli-gr4- tani (^-90°) . . . (I). 

sin | (H+ w) 

Now, |S. + ID = pKz = 90°— N ; 

whence, N = 90°- (IS + |D), 

or, 

N = 360° +90°- (|S + |D) = 180°-£S + 180°-iD + 90°; 

consequently, N = E + F + 90° . . . (J), 

rejecting 360° when the sum exceeds that number. 

Next, for the altitude of the nonagesimal, we have, (For. 88,) 

tan \h — pp-. tan | (H + w), 

cos |D 

1 tani(H+«)...(K). 



cos F* 

N and h being known, to obtain the formula for the parallax 
in longitude and latitude, we have only to replace in the formulae 

* Francceur's Uranography, p. 421. 
45 



354 APPENDIX. 

for the parallax in right ascension and declination, the altitude I of 
the pole of the equator by that 90°— h of the pole K of the eclip 
tic, and the distance im of the star s from the meridian by the dis- 
tance nc to the vertical through the nonagesimal. Let us change 
then in formulae (A), (B), (C), (D), (E), (F), and (G), I into 
90°— h, and q into fc — fn = L— N, L being the longitude fc of 
the star s. Besides, d will become the distance sK to the pole of 
the ecliptic, or complement of the latitude X = sc. Making these 
substitutions, and denoting the parallax in longitude by n, and the 
parallax in latitude by *, we obtain in terms of the apparent longi- 
tude and latitude, 

sinP sin^ / . T , T , > /T . 

sinll = — r— ; — sinL-N + n . • . (L), 
sina 

., . x sin (L— N + n) / sin P cos ^\ ,,,. 

coi(d + «)= — ^ — ==r- - y (coU r—r- ) . . . (M), 

sin(L— N) \ sina / 

sin P cos h - - 

tanz = r— -, — . . . (N), 

sina 

sin(L-N + n)c o s(<Z + g) 

cot(d+«)= V— ? '. * . . . (0), 

sm (L--N)smrfcosa? v ' 

sin ir = sin P cos h sin (d + *) ~"~ 

cos (d + <) cos (L— N + }n) sin P sin h 

cos in 

tan h cos (L— N + in) _ . 

tany= ^w ■••«). 

sin P cos h . . x . 

sin * = sin (a + tf — y) • • • (R); 

cos y 

and in terms of the true longitude and latitude, 

„ Psin/i . /T _ TX , /PsinAV 
n= . sm(L-N)+( ---r) x 
sin a \ sin a / 

sin(L-N)cos(L-N)sinl" . . . (S), 

PcosA . ,_ v . 7 /Pcos^\ 2 

* = sin (d — y) + i I I x 

cos y V cos y / 

sin 2 (d— y) sin 1" . . . (T), 

tan h cos (L— N+in) 

tan y = — ; '. 

9 cos |n 

To facilitate the computation, sin n, sin or, and sin P, in formu- 
lae (L), (P), and (R), maybe replaced by the arcs themselves. 

The distance d of the star from the pole of the ecliptic enters 
into these formulae in place of the latitude X. 

To find the apparent distance d', we have 
d' = d + « ; 



i'P), 



356 

for the apparent latitude X', 

V=X — *\ 

for the apparent longitude L', 

L'=L + n. 

The logarithmic formulae given on page 298, were derived from 
equations (L), (0), and (P), and the logarithmic formula on page 
299 from equa. (0). 

To determine now the effect of parallax upon the apparent di- 
ameter of the moon. 

Let ACB (Fig. 65, p. 147) represent the moon, and E the sta- 
tion of an observer j also let R = apparent semi-diameter of the 
moon, and D =ite distance. The triangle AES gives 

sin AES = ^r, or sin R = ^?r-. 

At any other distance D' we should have for the apparent semi- 
diameter R 7 , 

. AS 

sin K = -sry ; 

sin R' D 
whence, -r— ^- = =£. 

sin R D' 

Thus, if R' = moon's apparent semi-diameter to an observer at the 
earth's surface, as at (Fig, 26, p. 50), R =the same as it would 
be seen from the centre C, and S represents the situation of the 
moon. 

sin R' _ CS _ sin Z OS _ sin Z 
sin R OS sin ZCS sin z 
But we have, (see page 350,) 

smZ _(sinc? + &) sin (#+«) 
sin z sin d sin q 

or, in terms of the apparent longitude and latitude, (see page 354.) 
sin Z _ sin (d + *) sin (L — X + n) 
sin z sin d sin (L — X) 

„ 0J sin R sm id + ~) sin (L — N + IT) /TTX 

Hence, smR' = * — 7 ■ /T ™ \ . . (UV 

sm a sm(L — W) 

Aberration in Longitude and Latitude, and in Right Ascension 
and Declination* (See Art. 129, page 59.) 

Aberration is caused by the motion of light in conjunction with 
the motion of the earth. Light comes to us from the sun in 8 m - 
I7\8, during which time the earth describes an arc a =20' .44, 

* Francceur's Uranography, p. 442, 4tc 




356 .%FFENDIX. 

of its orbit pbdin (Fig, 127,) supposed circular: p is the place of 

the earth. Let us take any plane whatsoever, which we will call 

Fig. 127. relative, passing through the star and the 

^ sun, and let ddl be the intersection of this 

plane and the ecliptic, with which it makes 

an angle k : let us seek the quantity <p by 

/ jS \\ which the aberration displaces the star in 

tf ~~c^=r J \j' tne direction perpendicular to this plane. 

^x^y The question is to project on to a line per- 
pendicular to the relative plane, the small 
constant arc a which the earth describes, 
this being the quantity that the star is dis- 
placed from its line of direction in a direction parallel to the line 
of the earth's motion, (see Art. 124 of the text:) this projection 
is <p, variable according to the position of the relative plane in rela- 
tion to which it is estimated. The velocity along the tangent at 
p, makes with ph an angle 6 =pch = the arc pd' ; a cos 6 is then 
the projection of this velocity on the line ph. The angle of our 
two planes being k, this projection will be reduced to a cos 6 sin 
k, when it is taken perpendicularly to the relative plane. Thus, 
9 = a sin k cos 6 . . „ (V) . 
The aberration displaces the star from the relative plane by this 
quantity <p, k designating the inclination of this plane to the eclip- 
tic, and 6 the arc pd', reckoned from p the place of the earth to d' 
the point of intersection of these two planes. Let us give to the 
relative plane the positions which are met with in applications. 

Let us suppose at first that k — 90°, or sin k = 1 ; the relative 
plane will then be perpendicular to the ecliptic, Letn be the ver- 
nal equinox ; we have pd' = np — nd! ; np is the longitude of the 
^earth, or 180° + that O of the sun j nd' is the longitude I of the 
star ; whence 

<p = — a cos (O — 7). 
Fig. 128. Now, let M (Fig. 12S) be the true place 

^ V~. K °^ tne star ' ^' ^ e slar as displaced by 
aberration, KM is the circle oi true lati- 
tude, KM' the circle of apparent latitude, 
and MM' = 9 : this arc has its centre C 
on the axis which passes through the pole 
K of the ecliptic ; the longitude of the 
star is then altered by the part 00' of the 
ecliptic comprised between these twc 
planes j and since 00' is to the arc MM' 

as the radius 1 is to the radius CM = sin KM = cos latitude X of 

the star, we have 

aberr. in long. = cos (O — I) . . , (W), 

cos X 

If the relative plane is kc, (Fig. 129,) perpendicular to the circle 




ABERRATION IN RIGHT ASCENSION AND DECLINATION. 357 

•of latitude Kcd, the aberration 9 

perpendicularly to it, will be the 

aberration in latitude. Let kd be 

the ecliptic, and the earth ; the 

angle k is measured by the arc cd 

= X ; the arc ok = 6 = O — long. 

of k ; and as kd = 90°, long, of 

point k=l — 90° : substituting in equation (V), we find 

aberr. in lat = — a sin X sin (O — I) . . . (X). 

These aberrations of the star produce a small apparent orbit, 

which is confounded with its projection on the tangent plane to 

the celestial sphere. Let us suppose the orbit to be referred to 

two co-ordinate axes passing through the true place of the star and 

lying in the tangent plane, of which one is parallel to the plane of 

the ecliptic, and the other perpendicular to this, or tangent to the 

oc 

circle of latitude at the star : and let = aberr. in long., and 

cos X 

y = aberr. in lat. ; y will be the ordinate, and x (the aberr. in long., 

reduced to the parallel through the star) the abscissa : we have 

r- — — cos (G — l\ 

cos X cos X 

y = — a sin X sin ( O — I) ; 

at? 

or, — = — 00s ( O — 1% 

_4— =-saa(<D-*). 

<zsinX 

Squaring the last two equations, and adding them together, O dis- 
appears, and we find 

y 2 + z 2 sin 2 X=a 2 sin 2 X . . . (Y), 
whatever may be the place of the earth. Such is the equation of 
the apparent orbit, which, as we perceive, is an ellipse of which 
the semi-axes are a and a sinX, and whose centre is the true place 
of the star. "When the star is at the pole of the ecliptic, X = 90°, 
and the ellipse becomes a circle of which the radius is a. When 
X = •©., this ellipse is reduced to an arc 2a of the ecliptic. 

To find the aberration in right ascension, the relative plane must 
be perpendicular to the equator. Let kc be the equator, (Fig. 129,) 
p its pole, psd the relative plane, which is the circle of declination 
of the star s ; kd the ecliptic, the earth, k the vernal equinox, 
kc = R, sc=D. Aberration carries the star s out of the plane 
pcd a distance 0, which it is the question to determine. Equa. 
(V) is here 

<p = a sin d cos do = a sin d cos (kd — ko) 
= a sin d (cos kd cos ko -f- sin kd sin ko) 
= a sin d cos kd cos ko + a sin d sin kd sin ko 



358 . APPENDIX. 

but Jco =- long, of earth = 180° + O ; we have also the angle k = 
the obliquity u of the ecliptic, and the right-angled spherical trian- 
gle kcd gives, by Napier's rules, 

cot kd = cot R cos w, sin d sin kd = sin R, 

The 1st equa. multiplied by the 2d, gives 

sin d cos kd = cos R cos w, 

whence c?= — a (cos R cos u cos O + sir R sin O). 

The displacement from M to M'(Fig. 128) conducts, as before, 
to the division of 9 by cos D, to have the corresponding arc of the 
equator : thus the aberration in right ascension is, 

u = — a sin R sec D sin O — a cos w cos R sec D cos O (Z). 

Taking the relative plane perpendicular to the circle of declina- 
tion, we find for the aberration in declination, 

v = — a sin D cos R sin O — a cos u (tan u cos D — sin R sin D) 
cos O . . . -(a). 

These formulae may easily be adapted to logarithmic computa- 
tion : 

In formula (Z) let a sin R sec D = A, and a cos w cos R sec 
D = B ; then, 

u = - A (sin O +- r cos O) . . . (Z ; ). 

A. 

_ B a cos u cos R sec D _ /T . 

rut tan 9 = — = ■ : — = =^ — = cos w cot R . . . (o), 

A a sin R sec D 

and we shall have 

u = — A (sin O H r cos O) 

cos 9 

sin O cos 9 + sin 9 cos 

cos 9 

A • / 

sin (O + 9;. 



cos o 



Restoring the value of A, and taking 7c for sec D, we obtain 

& cos D 

a sin R . 

u = — -ye : sm(0 + o) . . . (c). 

cosD cos 9 v ' v ' 

The auxiliary arc tp is given by equation (b) ; it must be substi 
tuted in equation (c), with its sign, and we then obta n u. Tan 
9, and the co-efficient of sin ( O + 9) are constant, for the same star, 
for a long period of time, since these quantities vary very slowly 
with w and the precession. Moreover, the co-efficient of sin 
(O+9) is the maximum value of u, since it answers to sin 
(O +9) = 1- Thus we shall be able to calculate in advance, for 



NUTATION IN RIGHT ASCENSION AND DECLINATION. 359 

any designated star, the values of 9 and of the maximum of the aber- 
ration in right ascension, or of the logarithm of this maximum. 

The results of these calculations for 50 principal stars are given 
in Table XCI, columns entitled M and 9. 

If in equation (a) we make a sin D cos R = A', and a cos u 
(tan u cos D— sin R sin D) = B', we shall have the equation 

B' 

v =— A' (sin O + -77 cos G), 

in which A' and B' are constants. This equation is of the same 
form with equa.(Z'). We therefore have, in the same manner as 
for the right ascension, 

B' a cos u (tan w cos D — sin R sin D) 

tan & = -77 = — =r 15 

A' a sin D cos K 

_ a sin w cos D — a cos w sin R sin D 

a sin D cos R 

sin w cot D 



cos R 



— cos w tan R . . . (d\ 



A' . = , ' . a sin D cos R 

cos B v ' cos 6 

sin(O-M) . . , (e). 
6 is given by equation (d), and being substituted in equation (e) t 
we shall have v . B and the co-efficient of sin (O + d) are constant 
for the same star, and we can therefore calculate in advance the 
value of this arc, and of the co-efficient, which is the maximum 
of the aberration in declination. Columns entit]ed 6 and N, Table 
XCI, contain the quantities 6 and the logarithms of the maxima of 
the aberration in declination for 50 principal stars. 

For convenience in calculation, the angles 9, 6, and the maxima, 
M, N, in Table XCI, have been rendered positive in all cases. 
This has been accomplished by adding 12 s - to 9 and 6 whenever 
the calculation conducted to a negative value, and by adding 6 s - to 
O + 9, or O + 0, whenever the co-efficient had the sign — , (this 
sign being changed to + ;) in this manner the sign of each of the 
two factors is changed, which does not alter the sign of the pro- 
duct. 

Formula for the Nutation in Right Ascension and Declination.* 
(Referred to in Article 148, p. 63.) 

In deriving these formulae, we must begin with borrowing cer- 
tain results established by Physical Astronomy. It has been 
proved, in confirmation of Bradley's conjectures, that the phenom- 
ena of nutation are explicable on the hypothesis of the pole of the 
earth describing around its mean place (that place which, see page 

* Woodhouse's Astronomy, p. 357, &c. 



360 



APPENDIX. 



61, it would hold in the small circle described around the pole of 
the ecliptic, were there no inequality of precession) an ellipse, in 
a period equal to the revolution of the moon's nodes. The major 
axis of this ellipse is situated in the solstitial colure and equal to 
18". 50 ; it bears that proportion to the minor axis (such are the 
results of theory) which the cosine of the obliquity bears to the 
cosine of twice the obliquity : consequently, the minor axis will be 
13".77. 

Let CdA. (Fig. 130) represent such an ellipse, P being the mean 
place of the pole, K the pole of the ecliptic. CDOA is a circle 

Fig. 130. 
K 




described with the centre P and radius CP. VL is the ecliptic, 
Vw the equator, KPL the solstitial colure. In order to determine 
the true place of the pole, take the angle APO equal to the retro- 
gradation of the moon's ascending node from V : draw Oi perpen- 
dicular to PA, and the point in the ellipse, through which Oi 
passes, is the true place of the pole. This construction being ad- 
mitted, the nutations in right ascension and north polar distance 
may, Pp being very small, be thus easily computed. 



Nutation in North Polar Distance. 

Nutation in N. P. D. = Ptr— ptf. = Pr = Pp cospPtf, nearly, 
= P/>cos(APp-f APtf) 
= Pp cos ( APp + R — 90°) 
= Vp sin< APp + R), 

R denoting the right ascension. 



NUTATION IN RIGHT ASCENSION AND DECLINATION. 361 

Nutation in Right Ascension. 

The right ascension of the star <f is, by the effect of nutation, 
changed from Vw into Y'ts. Now, 

V'ts = Y'v + Yw + ts, nearly, 
whence, Vw — Y'ts = — Y'v — ts 

= - W cos TV'v - Pp sin Pptf ^"j 
^ ^ sinP^ 

in which expression Y'v (= VV cos VV'u) is, as in the case of pre- 
cession, common to all stars. 

In order to reduce farther the above expression, we have 
pP<s = APp + APtf = APp + R - 90°, 

^dYY' = Ll = Fp^-^-; 
r smPK 

whence, — V'v — ts = — Pp sin APp cot w 

- Pp sin (APp + R - 90°) cot N. P. D. 

= — Pp sin APp cot w + Pp cos (APp -f R) cot 5, 

<5 representing the north polar distance, and w the obliquity of the 
ecliptic. 

But these forms are not convenient for computation. In order 
to render them convenient, we must, from the properties of the el- 
lipse, deduce the values of Pp, and of the tangent of APp, and 
then substitute such values in the above expressions : thus, 
Pp __ sec APp _ cos APO _ cos (12 s — Q,) _ cos & 
PO sec APO cos APp cos APp cos APp* 

Q designating the longitude of the moon's ascending node ; 

, .p. PO cos Q 

whence Pp = r7 : — . 

cos APp 

tan APp _ pi _ Pe? _ Pd 
tanAPO _ Oi~PD _ PO ; 

hence, tan APp = p-~ tan APO = p-~ tan ( 1 2 s - — &) 

= ~PO 
Now substitute, and there will result 

The Nutation in North Polar Distance 

= — — - (sin APp cos R + cos APp sin R) 

cos APp 

= PO (tan APp cos R cos Q + cos & sin R) 
= — Pd cos R sin Q> -\- PO cos Q, sin R 
= - 6".887 cos R sin Q + 9".250 cos Q, sin R . . (/) ; 
46 



Again, 



362 APPENDIX. 

wliich is the difference, as far as nutation is concerned, between 
the mean and apparent north polar distance. The apparent north 
polar distance, therefore, must be had by adding the preceding 
quantity, with its sign changed, to the mean. 

Nutation in right ascension = Yd sin Q> cot w 
+ PO cos Q cos R cot o + Yd sin Q> sin R cot S, 
which, as far as nutation is concerned, is the difference of the mean 
and apparent right ascensions : and, consequently, the above ex- 
pression must be subtracted from the mean, in order to obtain the 
apparent right ascension ; or, which is the same, must be added 
after a negative sign has been prefixed ; in which case, we have, 
substituting for PO, Yd their numerical values, 

The Nutation in Right Ascension 

= — 6". 887 sin & cot w 

— 9".250 cos & cos R cot 8- 6". 887 sin Q, sin R cot 5 . . . (g). 

Formulae (/) and (g) are of the same form with (Z) and {a) for 
the aberrations in right ascension and declination, and therefore 
formulae may be derived from them similar to (c) and (e), adapted 
to logarithmic computation. The quantities corresponding to <p, 
M, £, N, have been calculated for the stars in the catalogue of 
Table XC, and inserted in Table XCI, in the columns entitled 
?'. M', 6', N'. 

The Solar Nutation arises from like causes as the Lunar, and 
admits of similar formulae. As an ellipse, made the locus of the 
true place of the pole, served to exhibit the effects of the lunar 
nutation, so an ellipse, of different, and much smaller dimensions, 
may be made to represent the path which the true pole of the 
equator would, by reason of the sun's inequality of force in caus- 
ing precession, describe about the mean place of the pole. Thus, 
in Figure 130, the ellipse AdC will serve to represent the locus 
of the pole, when AP = 0".545, Yd = 0".500, and APO, instead 
of being = Q, is equal to 2 0, or twice the sun's longitude, 
taken in the order of the signs ; the equations, therefore, for the 
solar nutation in north polar distance, and right ascension, analo- 
gous to eqs./and g will be 

The Solar Nutation in North Polar Distance 

= - 0".500 cos R sin 2 © + 0".545 sin R cos 2 . . . (h). 

The Solar Nutation in Right Ascension 

= — 0".500 sin 2 cot w 

— 0".545 cos 2 cos R cot 6 — 0".500 sin 2 sin R cot 6 . . (t). 

If the apparent place of a star should be required with great 
precision, it would be necessary to compute the solar nutations 
from these formulas, and apply them as corrections to the mean 



EFFECTS OF OBLATENESS OF THE EARTH'S SURFACE. 363 

right ascension and declination. The calculation would be per- 
formed after the same maimer as for the lunar nutation ; but it is 
much abridged by remarking that the form of the equations is the 
same as that of the equations for the lunar nutation, and that the 
co-efficients are very nearly the 0.075 of those of the latter equa- 
tions. Thus we can make use of the same arcs <p', &', and log. 
maximay M', N', repeat the calculation for the lunar nutation, 
taking 2 O instead of &>, and multiply the nutations in right ascen- 
sion and declination thus obtained by 0.075. The results will be 
the solar nutations required. (See Prob. XX.) 



F rrmulcz for computing the effects of the Oblateness of the Earth? s 
Surface upon the Apparent Zenith Distance and Azimuth of a 
Star* (See Article 162, page 69.) 

From the centre of the earth, an observer would see a star at I, 
Yig. i3i. (Fig. 131,) and would have V for his 

zenith : from the surface his zenith is 
Z, and he sees this star at B ; IB* —p 
is the parallax in altitude ; the azi- 
p muth VZI is changed into VZB. If 
for a given time, we wish to calculate 
the apparent zenith distance BZ, and 
the apparent azimuth VZB, we have 
first to resolve the spherical triangle IZP, in which we know the 
two sides ZP = co-latitude and IP — co-declination, and the in- 
cluded hour angle P ; the azimuth VZI (= A), and the arc IZ 
(= n) will thus be known. But from the earth's surface, the star 
is seen at B : the azimuth VZB — VZI -fTZB = A -f- a ; the zenith 
distance BZ = n -\-p, since, VZ (= i) being very small, we have 
sensibly IB + IZ = BZ. By reason of the want of sphericity of 
the earth, parallax then increases the true azimuth and zenith 
distance of a star by small quantities, a and p, which it is neces- 
sary to calculate. In the triangle VIZ we have 

cos IV = cos i cos n -f- sin i sin n cos A == cos n -f- k sin n ; 

making cos i = 1, sin i = i, and i cos A = k. Now, k L i, and 
a fortiori cos k = 1, sin k = k ; whence 

cos IV = cos n cos k + sin n sin k = cos (n — k), 
and IV = n — k =n — i cos A. 

Thus we correct the calculated arc n by the quantity — i cos 
A, to have 

IV = z = n — i cos A . . . (J). 
If this value of z be introduced into equation (10), page 52, we 

* Francoeur's Uranography, p. 426, &c 




364 



APPENDIX. 



shall have p, and thence the apparent zenith distance Z = n +p 
= BZ. 

Afterwards, to obtain IZB = a, or the parallax in azimuth, the 
triangles ZBV, ZBI give 

sin ZBV _ sin (A + a) sin ZBV ^sina. 
sin i sin (z -\-p) ' sin n sinp ' 

whence, by equating the values of sin ZBV, 

sin n sin a _ sin i sin (A + a) ^ 
sin^> sin (z -f p) 

substituting for sin p its value sin H sin (z +p) = sin H sin Z, 
(equa. 8, page 51,) and reducing, we have 

sin a _ sin (A + a) 



sin H sin i 



sinn 



and as i is very small, sin i sin (A + a) does not differ sensibly 
from i sin A, and we thus have in seconds, (For. 47, page 343,) 

Hi sin A sin 1" /7X 



sin n 



Solution of Kepler's Problem, by which a Body's Place is found 
in an Elliptical Orbit* (See Art. 268, p. 106.) 

Let APB (Fig. 132) be an ellipse, E the focus occupied by the 
sun, round which P the earth or any other planet is supposed to 
revolve. Let the time and planet's motion be dated from the ap- 

Fig. 132. 

M 




side or aphelion A. The condition given is the time elapsed from 
the planet's quitting A ; the result sought is the place P ; to be 
determined either by finding the value of the angle AEP, or by 

* Woodhouse's Astronomy, p. 457, &c. 



solution of kepler's problem. 365 

cutting off, from the whole ellipse, an area AEP bearing the same 
proportion to the area of the ellipse which the given time bears to 
the periodic time. 

There are some technical terms used in this problem which we 
will now explain. 

Let a circle AMB be described on AB as its diameter, and sup- 
pose a point to describe this circle uniformly, and the whole of it 
in the same time as the planet describes the ellipse ; let also t de- 
note the time elapsed during P's motion from A to P ; then if AM = 

— :— i x 2 AMB, M will be the place of the point that moves 
period r 

uniformly, while P is that of the planet; the angle ACM is 
called the Mean Anomaly, and the angle AEP is called the True 
Anomaly. 

Hence, since the time (t) being given, the angle ACM can al- 
ways be immediately found, (see Art. 267, p. 106,) we may vary 
the enunciation of Kepler's problem, and state its object to be the 
finding of the true anomaly in terms of the mean. 

Besides the mean and true anomalies, there is a third called the 
Eccentric Anomaly , which is expounded by the angle DC A, and 
which is always to be found (geometrically) by producing the ordi- 
nate NP of the ellipse to the circumference of the circle. This 
eccentric anomaly has been devised by mathematicians for the 
purposes of expediting calculation. It holds a mean place between 
the two other anomalies, and mathematically connects them. There 
is one equation by which the mean anomaly is expressed in terms 
of the eccentric ; and another equation by which the true anomaly 
is expressed in terms of the eccentric. 

We will now deduce the two equations by which the eccentric 
is expressed, respectively, in terms of the true and mean anomalies. 
Let t = time of describing, AP, 

P = periodic time in the ellipse, 

a =CA, 

ae=EC, 

v = L PEA, 

u = L DCA ; (whence, ET, perpendicular to DT, = EC 
x sin u,) 

P =PE, 

-r = 3.14159, &c. ; 
then, by Kepler's law of the equable description of areas, 

« = P x J^L =P x5££iDEA P 
area oi ellip. area circle *ar 

P /ET.DC AD.DC\ Pa,™ , ^ , 

P Pi 

= -rr- (e sin u + u) : hence, if we put-— =-, 



366 APPENDIX. 

we have 

nt = e sin u + u. . . . (7), 

an equation connecting the mean anomaly nt, and the eccentric u, 
In order to find the other equation, that subsists between the 
true and eccentric anomaly, we must investigate, and equate, two 
values of the radius-vector p, or EP. 

First value of p, in terms of v the true anomaly, 

o(l_-^ ( 

1—ecosv 

Second, in terms of u the eccentric anomaly, 

p = a (1 + e cos u) . . . (2). 
For, p 2 = EN 2 + PN 2 

= EN 2 + DN 2 x (1 - e 2 ) 

= (ae + a cos uf + « 2 sin 2 u (1 — e 2 ) 

= a 2 je 2 + 2e cos m + cos 2 m| + a 2 (1 — e 2 ) sin 2 m 

= a 2 \ 1 -f- 2e cos m + e 2 cos 2 m } . 

Hence, extracting the square root, 

p = a (1 +e cos w). 

Equating the expressions (1), (2), we have 

(1 — e 2 ) = (1 — e cos v) (1 + e cos u), whence, 

e -f- cos u . . 

cos v = - — ; , an expression for v in terms of u : 

1 + e cos u r 

but, in order to obtain a formula fitted to logarithmic computation, 

v 
we must find an expression for tan - : now, (see For. 12, p. 341,) 

v _ //l — cos v\ _ /({I — e ) (1 — cos u )\ 
tan 2 " V Vl +cos~i/ " = V V(l +e)(l +cosm)/ 

-VtriJ)^ ■;"■(»>■ 

These two expressions (/) and (w), that is, 
nt = e sin u + w, 
i; //l — e\ m 

tan 2 = v vmr an 2' 

analytically resolve the problem, and, from such expressions, by 
certain formulae belonging to the higher branches of analysis, may 
v be expressed in the terms of a series involving nt. 

Instead, however, of this exact but operose and abstruse method 
of solution, we shall now give an approximate method of express- 
ing the true anomaly in terms of the mean. 

MO is drawn parallel to DC. (1.) Find the half difference of 



solution of kepler's problem. 367 

the angles at the base EM of the triangle ECM, from this ex- 
pression, 

tan i (CEM - CME) = tan \ (CEM + CME) x \^ i 

in which, CEM -f CME = ACM, the mean anomaly. 

(2.) Find CEM by adding \ (CEM + CME) and \ (CEM 

— CME) and use this angle as an approximate value to the ec- 
centric anomaly DC A, from which, however, it really differs by 
L EMO. 

(3.) Use this approximate value of L DCA == L ECT in 
computing ET which equals the arc DM ; for, since (see p. 365), 
P 

t = : — r- x DE A, and (the body being supposed to revolve in 

area circle 

P 

the circle ADM) = — r x ACM, area AED = area ACM, 

area circle 

or, the area DEC + area ACD — area DCM + area ACD ; con- 
sequently the area DEC =the area DCM, and, expressing their 
values, 

ET x DC DM x DC , , _„_ ^ 

= , and thus, ET = DM. 

Having then computed ET = DM, find the sine of the resulting 
arc DM, which sine = OT ; the difference of the arc and sine 
(ET - OT) gives EO. 

(4.) Use EO in computing the angle EMO, the real difference 
between the eccentric anomaly DCA and the L MEC ; add the 
computed L EMO to L MEC, in order to obtain L DCA. The 
result, however, is not the exact value of L DCA, since L EMO 
has been computed only approximately; that is, by a process which 
commenced by assuming L MEC for the value of the L DCA. 

For the purpose of finding the eccentric anomaly, this is the 
entire description of the process ; which, if greater accuracy be 
required, must be repeated ; that is, from the last found value of 
L DCA = L ECT, ET, EO, and L EMO must be again com 
puted. 



369 



NOTE I. 

The number of planets known at the present date (January 1st, 1852), is 
twenty-two. During the last seven years twelve new planets have been dis- 
covered. The following table contains the names of these planets, together 
with the date and place of discovery, and the name of the discoverer. 



Names. 


"When discovered. 


By whom. 


Where. 


Astrsea 


Dec. 8, 1845 


Hencke 


Driessen. 


Neptune 


Sept, 23, 1846 


Galle 


Berlin. 


Hebe 


July 1, 1847 


Hencke 


Driessen. 


Iris 


Aug. 13, 1847 


Hind 


London. 


Flora 


Oct. 18, 1847 


Hind 


London. 


Metis 


April 25, 1848 


Graham 


Markree. 


Hygeia 


April 12, 1849 


Gasparis 


Naples. 


Parthenope 


May 13, 1850 


Gasparis 


Naples. 


Clio 


Sept, 13, 1850 


Hind 


London. 


Egeria 


Nov. 2, 1850 


Gasparis 


Naples. 


Irene 


May 20, 1851 


Hind 


London. 


Eunomia 


July 29, 1851 


Gasparis 


Naples. 



Although Neptune was first seen by Galle, at Berlin, the honor of the discov- 
ery of this planet is generally awarded to Leverrier, a French astronomer. 
Leverrier ascertained, from a careful examination of the motions of Uranus, that 
that planet must be subject to the disturbing action of an unknown planet more 
remote from the sun. He investigated the probable orbit and mass of this 
unknown planet, that is, the orbit and mass that would serve to account for the 
previously unexplained irregularities observed in the motions of Uranus, and 
assigned its probable place in the heavens. At his request Galle, of the Berlin 
Observatory, undertook the search for it ; and on directing his telescope to the 
part of the heavens designated by Leverrier, detected the supposed planet 
within 1° of the place which had been assigned by that astronomer. 

The same investigation was undertaken about the same time, and with very 
nearly the same results, by a young English mathematician by the name of 
Adams, who is therefore entitled to a share of the honor of this wonderful 
discovery. 

The planets Ceres, Pallas, Juno, and Vesta, on account of their diminutive 
size and certain other peculiarities, have received the appellation of Asteroids. 
All the newly-discovered planets, with the exception of Neptune, are also classed 
among the asteroids. The number of asteroids at present known is, accordingly, 
fifteen. " Besides these fifteen, others yet undiscovered may exist ; and it is 
extremely probable that such is the case, — the multitude of telescopic stars 
being so great that only a small fraction of their number has been sufficiently 
noticed to ascertain whether they retain the same place or not," and from one 
to three new asteroids having been discovered every year since 1846. 

47 



370 



notes n. ni. 



NOTE II. 

At the present date (Jan., 1852), the largest and best telescope in the United 
States is the great refractor at the Cambridge Observatory, manufactured by 
Merz and Mahler, of Munich, Bavaria. The aperture of the object-glass is 15 
inches, and its focal length is 22£ feet. It has 18 different powers, varying from 
180 to 2,000. Its dimensions are a trifle greater than those of the Pulkova 
refractor, and it is generally conceded to be superior to it in its performance. 
It is, accordingly, the best refracting telescope in the world. It was erected in 
June, 184*7, and in the hands of Messrs. "W". C. and G-. P. Bond has already 
enriched astronomy with many valuable observations and discoveries. 

The accuracy of transit observations has recently been greatly increased by 
the introduction of the Electro-Chronograph ; by which, with the adaptation of a 
proper electro-magnetic recording apparatus, the seconds measured off by the pen- 
dulum of a clock are designated by a series of equally distant dots or breaks in a 
continuous line, upon a fillet or roll of paper to which an equable motion is given 
by machinery. The observer holds in his hand a break-circuit key, by means of 
which he interrupts the circuit at the instant that the star is bisected by one of 
the wires in the field of the telescope, and thus makes a break in one of the 
short lines on the fillet, that designate the duration of the successive seconds. 
In this way it is believed that the instant of the transit across a single wire can 
be noted to within a much smaller fraction of a second than by the common 
method. Besides, the number of bisections in a single culmination of a star, by 
increasing the number of wires, may be multiplied some seven-fold. 

This method of observation has been introduced at the Cambridge Observa- 
tory, and also at the National Observatory. 



NOTE III. 



ELEMENTS OF THE OEBITS OF THE ASTEROIDS, 
Arranged in the Order of their Mean Distance from the Sun. 





Name. 


Distance. 


Period in 
days. 


Eccentricity. 


Inclination. 


Longitude of 
Ascending Node. 


1 


Flora 


2.201687 


1193.249 


.156557 


o / // 
5 43 4.8 


o / ;/ 
110 18 12.0 


2 


Clio 


2.334876 


1303.127 


.217922 


8 23 1.9 


235 19 49.8 


3 


Vesta 


2.361081 


1325.147 


.089569 


7 8 29.7 


103 23 31.6 


4 


Iris 


2.380624 


1341.636 


.229942 


5 28 15.9 


259 48 10.2 


5 


Metis 


2.385607 


1345.850 


.120253 


5 34 27.8 


68 32 17.4 


6 


Eunomia 


2.399440 


1357.573 


.136504 


13 18.5 


292 51 1.8 


7 


Hebe 


2.425786 


1379.994 


.200180 


14 47 56.0 


138 29 42.6 


8 


Parthenope 


2.450833 


1401.000 


.099466 


4 36 56.7 


124 57 55.8 


9 


Irene 


2.552303 


1518.943 


.170022 


8 37 35.7 


87 47 46.2 


10 


Egeria 


2.560070 


1492.230 


.096180 


15 57 59.8 


43 35 24.4 


11 


Astraea 


2.577047 


1511.095 


.188058 


5 19 22.7 


141 25 14.6 


12 


Juno 


2.670837 


1594.296 


.254884 


13 3 22.1 


170 54 45.6 


13 


Ceres 


2.768051 


1682.125 


.076652 


10 37 4.4 


80 48 66.6 


14 


Pallas 


2.772858 


1686.510 


.239815 


34 37 33.0 


172 43 59.7 


I 15 


Hygeia 


3.150060 


2042.101 


.010103 


3 47 15.5 


287 37 8.6 



NOTE IV, 



371 





Name. 


Longitude of 
Perihelion. 


Mean Anomaly 
at Epoch. 


Epoch in Mean Time. 






o / // 


O / -4-4 




d, h. 


1 


Flora 


33 40.8 


35 48 7.0 


Berlin M. T 


1848, Jan. 1 


2 


Clio 


302 55 1.5 


65 47 23. 


« 


1851, Jan. 


3 


Vesta 


250 46 32.2 


225 44 18.8 


« 


1850, Jan. 9 


4 


Iris 


41 41 13.5 


330 41 54. 


■H 


1848, Jaa 1 0' 


5 


Metis 


70 33 42.8 


146 30 18. « 


u 


1848, May 5 12 


6 


Eunomia 


112 18 15.6 


172 10 21.6 


a 


1851, Aug. 5 


7 


Hebe 


14 50 50.3 


275 8 51.3 


" 


1847, Jan, 1 


8 


Parthenope 


316 49 51.8 


288 40 43.2 


<i 


1850, May 25 


9 


Irene 


191 8 27.5 


41 57 9.5 


Greenwich 


1851, June 10 


10 


Egeria 


116 26 49.4 


288 37 17. 


a 


1850, Nov. 2 


11 


Astrsea 


135 20 47. 


318 45 3.3 


Berlin 


1846, Jan. 1 


12 


Juno 


54 24 12.8 


124 31 10.8 


" 


1850, April 8 


13 


Ceres 


147 46 12.4 


219 6 29.5 


u 


1850, Sept. 25 Q 


14 


Pallas 


121 21 48.5 


217 31 10.6 


« 


1850, Aug. 23 


15 


Hygeia 


227 49 54.2 


330 52 8.5 


" 


1849, April 15 



NOTE IV. 



The number of planets which are now known to have the situations men- 
tioned in the text is no less than fifteen. It is a remarkable fact, with respect 
to these asteroids, as they are called, that their orbits, if we except those of 
Iris and Hygeia, have approximately two common points of reunion in opposite 
regions of the heavens. This singular fact is in accordance with a theory pro- 
pounded by Dr. Olbers nearly fifty years ago (1802), after the discovery of 
Ceres and Pallas, that " these small bodies were merely the fragments of a larger 
planet, which had exploded from some internal convulsion, and that several more 
might yet be discovered." For, since the supposed fragments must have origi- 
nally diverged from the same point, their paths must, agreeably to the laws of 
planetary motions, have two common points of reunion ; viz., the place occupied 
by the primitive planet at the time when the convulsion occurred, and the point 
in the heavens diametrically opposite to this. It is true that, as a matter of 
fact, the intersection is only approximate, the deviations from a common point 
being in some instances as much as 4°, and in the case of the planetoids Iris and 
Hygeia no less than 9°, but this discrepancy is ascribed, fey the advocates of 
Olber's theory, to the disturbing actions of the planets, and the consequent sec- 
ular displacement of the orbits of the asteroids, and it is accordingly conjectured 
that if the secular motion of the node of each orbit were known, we might, by 
■calculating back, find that at some period in the past the orbits all had truly a 
■common point of intersection, and thus determine the date of the supposed ex- 
plosion of the single primeval planet. On this point Professor Loomis remarks 
that " we may safely assume that the nodes of all the asteroids have not coin- 
cided within a period of many thousand years ; and therefore that, if these bodies 
aare the fragments of a larger planet which has exploded, this explosion must 
>aave taken place at a very remote epoch. 

" It should also be observed, that not only must the nodes of all the asteroids 
coincide, but the distance of the planets from the sun must be the same at that 
instant Now the distance of these planets from the sun when at their nodes, 
varies by nearly the radius of the earth's orbit ; so that to bring them all to- 
gether, we must suppose a corresponding change in the place of their perihe- 
lia. This also would require the lapse of many centuries ; and when we con- 
sider the necessity of a coincidence at the same instant, both in distance and di- 
rection, we can easily suppose that such a, result could not have taken place 
within a million of years." 



372 note v. 

ffOTE V. 

Gambart's or Biela's comet, at its return in 1846, exhibited a phenomenon al= 
together unprecedented in the annals of astronomy. On the 13th of January, at 
the National Observatory in "Washington, and on the 15th and subsequently, at 
all the principal observatories in this country and Europe it -was distinctly seen 
to have become double ; a very small and faint cometic body r having a nucleus of 
its own, being observed appended to it at a distance of about 2' from its centre. 
The two comets moved on side by side, for a period of two months, and through 
an arc of more than- 10°, when the companion, after undergoing remarkable 
changes of magnitude and luminosity, disappeared. During the whole of this 
interval the apparent distance between the two bodies gradually increased, but 
the apparent direction of the line of junction remained nearly the same. On' 
the 30th of January, the distance of separation had increased to 3 Von the 13th 
of February to &', and so until on the 5th of March it was over 9'. " Both bodies 
had nuclei, both had short tails, parallel in direction, and nearly perpendicular 
to the line of junction; but whereas, at its first observation on January 13th, 
the new comet was extremely small and faint, in comparison with the old, the 
difference, both in point of light and apparent magnitude, diminished. On the 
10th of February, they were nearly equal, although the day before the moon- 
light had effaced ,the new one, leaving the other bright enough to be well ob- 
served. On the 14th and 16th, however, the new comet had gained a decided 
superiority of light over the old, presenting at the same time a sharp and star- 
like nucleus, compared by Lieut. Maury to a diamond spark. But this state of 
tilings was not to continue. Already, on the 1 8th, the old comet had regained 
its superiority, being nearly twice as bright as its companion, and offering an un- 
usually bright and starlike nucleus. From this period the new companion began 
to fade away," but continued visible until after the middle of March. As seen 
by the author on the 17th of March in a reflecting telescope of 14 ft. focus, 
with a low power, the cometic mass had two points of maximum brightness, but 
the twin comets were not distinctly separate, On March 21 it appeared in the 
same telescope as one nebulous mass, with a single point of concentration. On 
the 22d of April this had disappeared. 

" While this singular interchange of light was going forward, indications of 
some sort of communication between the comets were exhibited. The new or 
companion comet, besides its tail, extending in a direction parallel to that of the 
other, threw out a faint arc of light which extended as a kind of bridge from 
the one to the other ; and after the restoration of the original comet to its former 
pre-eminence, it, on its part, threw forth additional rays, so as to present (on the 
22d and 23d of February, as seen by Lieut. Maury, of the National Observatory) 
the appearance of a comet with three faint tails forming' angles of about 120° 
with each other, one of which extended towards its companion." 

What was the relation of these two bodies ? Was the original comet actually 
divided into two, as appearances seemed to indicate ? Professor Plantamour r 
director of the observatory of Geneva, has furnished a partial answer to these 
questions. He has found that all the observations are very well represented 
by supposing that each nucleus described an independent ellipse around the 
sun. He has computed the. orbits of the two bodies upon this supposition, from 
the extensive and careful series of observations made upon them, and taking 
into account the disturbing influence of Jupiter, Mars, the Earth, and Venus- 
and concludes that " the disturbing action of one nucleus upon the other must 
have been extremely small, and that it is doubtful whether the observations 
were sufficiently precise to render this influence in any degree sensible. He 
has also shown that the increase of distance between the two nuclei, at least 
during the interval from February 10th to March 22, was simply apparent, be- 
ing due to the variation of distance from the earth and to the angle under which 
their fine of junction presented itself to the visual ray ; the real distance during 
all that interval (neglecting small fractions) having been on an average about 
thirty -nine times the semi-diameter of the earth, or less than two-thirds the dis- 
tance of the moon from the earth's centre." 

If it be true that the two bodies are in no sensible degree disturbed by their 
mutual actions, as M, Plantamour infers from his investigations, and as we should 



IfTOTES VI-IX. 373 

-uuvuaih suppose from the pTobable minuteness of the two cometary masses, It 
Jias been ealcukled by Sir John Herschel, from Plantamour's elements, that 
there will be an interval of 16 d - 4 between their next perihelion passages; " and 
it will be therefore necessary, at their next reappearance, to look out for each 
comet as a separate and independent body." "Nevertheless," as remarked by 
Herschel. " as it is still perfectly possible that some link of connection may sub- 
sist between them, it will not be advisable to rely on this calculation to the 
neglect of a meet vigilant search throughout the whole neighborhood of the 
more conspicuous one, lest the opportunity should be lost of pursuing to its con- 
clusion the history of this strange occurrence." 

The investigations of M. Plantamour have served to establish that the actual 
separation of the two bodies did not occur at the time of the apparent separa- 
tion in 1846- At what point of time anterior to that epoch it took place, it 
would seem to be impossible to determine. In fact, it is quite possible that the 
two bodies have been revolving independently of each other for an indefinite 
time, and that the supposed division of one comet into two was really the chance 
approach of two independent cometary bodies. Plantamour remarks that " the 
^extraordinary changes which the companion exhibited within the period of a 
few days, and which have often been noticed in other comets, seem to indicate 
that the brightness of these objects does not depend merely upon their distance 
from the earth and sun, but upon other unknown causes. These causes might 
have developed sufficient brightness in the companion at its*late return to the 
sun to render it visible to us ; while at its former returns, on account of its un- 
favorable position, the companion was too faint to be noticed/' 



NOTE VI.. 



The list given in the text has recently been increased by the addition of several 
other comets, viz., De Vico's comet, period t\ years, perihelion passage Sept. 2d, 
1844 ; Brorsen's comet, period, according to Hind, §\ years, perihelion passage 
Feb. 25th, 1847 ; Peters' comet, period nearly 16 years, perihelion passage June 
let, 1846. 



NOTE VII. 

The reader will find a complete catalogue of all comets whose orbits have been 
'determined, up, to 1846, in the American Almanac for 1841. 



NOTE VIII 

The new planet, Neptune, proves to be the third planet in the order of mag- 
caitude, being a little larger than Uranus. The newly-discovered .asteroids 
.are probably of a more diminutive size than the other four. 



NOTE IX. 



A remarkable analogy in the periods of rotation of the primary planets was 
•discovered a few years since (1848) by Daniel Kirkwood, of Pottsville, Pennsyl- 
vania. This analogy is now generally known by the name of Kirkwood's Law, 
juid is as follows : 

" Let P be the point <of equal attraction between any planet and the one next 



374 note x. 

interior, the two being in conjunction : P' that between the same and the one? 
next exterior. 

Let also D = the sum of the distances of the points P,. P' from the orbit of the 
planet ; which I shall call the diameter of the sphere of the planet's attraction ; 

D / =the diameter of any other planet's sphere of attraction found in like 
manner ; 

n = the number of sidereal rotations peformed by the former during one side- 
real revolution round the sun ; 

w / = the number performed by the latter : then it will be found that 



D 3 : D* 3 : or n 



w '(f)" 



That is, the square of the number of rotations made by a planet during one revo- 
lution round the sun, is proportional to the cube of the diameter of its sphere of 

attraction ; or — is a constant quantity for all the planets of the solar system. 

The analogy thus announced has been subjected to a rigid mathematical ex- 
amination by Mr. Sears C. Walker, with the following result : " "We may there- 
fore conclude," says he r " that whether Kirkwood's Analogy is or is not the ex- 
pression of a physical law, it is at least that of a physical fact in the mechanism 
of the universe." • (See the American Journal of Science, New Series, vol. x. 
pp. 19-26.) 

There are but three planets, viz., Venus, tte Earth, and Saturn^ for which all 
the elements embraced in this law are known. The diameters of the spheres of 
attraction of Mercury and Neptune are, from the nature of the case, incapable 
of determination. The mass of the one planet into winch the asteroids are sup- 
posed once to have been united is not known with certainty, as there may be 
asteroids yet undiscovered, and its period of rotation is hypothetical only. The 
diameters of the spheres of attraction of Mars and Jupiter can only be approx- 
imately determined ; and the period of rotation of Uranus is unknown. Pro- 
fessor Loomis, in a recent article, argues with a good deal of plausibility, that 
u Uranus and the asteroids cannot be reconciled with Kirkwood's Law by any 
admissible assumption with regard to the value of their elements." (See Sill!- 
man's Journal,, vol. xi. p. 217.) 

The objections urged by Professor Loomis have been answered by Professor 
Kirkwood. (See the Journal of Science, Second Series, vol. xi. p. 394.) The 
considerations adduced by him have served materially to weaken the force of 
these objections. 

The interest naturally awakened by the announcement of so important a dis- 
covery was heightened by the fact, that it was at once perceived that it furnished 
a new and powerful argument ic support of the nebular hypothesis (or cosmog- 
ony) devised by Laplace. (See a paper on this subject by Dr.. K A. Gould, Jr., 
in the Journal of Science, New Series, vol. x p. 26,. &c.) 



NOTE X. 



A new ring of Saturn, interior to the other two, was discovered by Mr. G-. Fl 
Bond, assistant at the observatory of Harvard University, on the 11th of 
November,. 1850. It was subsequently observed by the Messrs. Bond on re- 
peated occasions, from that date to the 7th of January, 1851. It shone with a 
pale dusky light. Its inner edge was sharply defined, but the side next the old 
ring was not so definite ; so that it was impossible to make out with certainty 
whether the new was connected with the old ring- or not. According to Mr. 
Bond's measurements the breadth of the new ring is l'\5. 

" The same appearances were noticed by the Rev. W. R. Dawes, at his ob- 
servatory, near Maidstone, in England, on the 25th and 29th of November, and 
subsequently by Mr. Lassell, of Starfield, near Liverpool." 



:notes xi, xn. 375 

Mr. G. P. Bond has propounded a bold and ingenious theory relative to the 
physical constitution of Saturn's rings; which is, that." they are in a fluid state, 
and -within certain limits change their form and position in obedience to the 
laws of equilibrium of rotating bodies." He conceives, also, that under peculiar 
circumstances of disturbance several subdivisions of the two fluid rings may 
take place, and continue for a short time until the sources of disturbance are 
removed, when the parts thrown off would again reunite. He supports his 
theory by arguments drawn from the results of observation, and by certain 
physical considerations. The chief argument derived from observation is, that 
several apparent subdivisions of the double ring have been noticed by different 
observers from time to time, and that these have in general been invisible to the 
same observers with the same telescopes, and under equally favorable circum- 
stances, and have also entirely escaped the observation of many other observers 
provided with equally good telescopes. It is supposed that these facts admit 
of explanation only on the hypothesis that the ring is a fluid mass, capable of 
occasional subdivision. (See Mr. Bond's original paper on this subject, published 
in Nos. 25 and 26 of the Astronomical Journal.) 

Professor Peirce, of Harvard University, has followed up the speculations of 
Mi*. Bond, by undertaking to demonstrate, from purely mechanical considera- 
tions, that Saturn's ring cannot be solid. " I maintain, unconditionally," says he, 
" that there is no conceivable form of irregularity and no combination of irregu- 
larities, consistent with an actual ring, which would serve to retain it perma- 
nently about the primary if it were solid." 

He is led by his investigations to the curious result, that Saturn's ring is 
sustained in a position of stable equilibrium about the planet solely by the 
attractive power of his satellites ; and that " no planet can have a ring unless it 
is surrounded by a sufficient number of properly arranged satellites." (See 
Astronomical Journal for June 16th, 1851.) 



NOTE XI. 



The seventh satellite of Saturn, in the order of distance from the primary, 
was discovered by the Messrs. Bond, with the great refractor of the Cambridge 
Observatory, on the 16th of September, 1848; and observed two days after- 
wards by Mr. Lassell, at Starfield, near Liverpool, with his large reflector. In 
fact, it appears to have been distinctly made out to be a satellite by these two 
observers on the same night, viz., that of the 19th of September. 

" The orbit of the new satellite serves to fill up a large chasm before existing 
between the 6th and 8th satellites (see Table VI). It is fainter than either of the 
two interior satellites discovered by Sir "William Herschel. Its time of revolu- 
tion is about 21.18 days, the semi-axis of its orbit, at the mean distance of 
Saturn, 214", and Messrs. Bond and Lassell have concurred in giving it the 
name of Hyperion." 

The periods of revolution, and the mean distances of the satellites of Saturn 
from their primary, together with the mythological names proposed for them by 
Sir John Herschel, are given in Table VI. 



NOTE XII. 

"Two of the satellites of Uranus are much more conspicuous than the rest, 
and their periods and distances from the planet have been ascertained with 
tolerable certainty. They are the second and fourth of those set down in the 
synoptic table (Table VI). Of the remaining four, whose existence, though an- 
nounced with considerable confidence by their original discoverer, could hardly 
be regarded as fully demonstrated, two only have been hitherto re-observed ; 



376 note xra. 

viz., the first of our table, interior to the two larger ones, by the independent 
observations of Mr. LasselL and M. Otto Struve, and the third, intermediate be- 
tween the larger ones, by the former of these astronomers. The remaining two, 
if future observation should satisfactorily establish their real existence, will 
probably be found to revolve in orbits exterior to all these." (Herschel's Out- 
lines of Astronomy, Art. 551.) 

It is just announced (Nov. 28th, 1851), that Mr. Lassell has discovered two 
new satellites attending upon Uranus. The following information is communi- 
cated with respect to them: "They are interior to the innermost of the two 
bright satellites first discovered by Sir "William Herschel, and generally known 
as the second and fourth. It would appear that they are also interior to Sir 
"William's first satellite, to which he assigned a period of revolution of about five 
days and twenty-one hours, but which satellite I have as yet been unable to 
recognize. I first saw these two of which I now communicate the discovery, on 
the 24th of last month, and had then little doubt that they would prove satel- 
lites. I obtained further observations of them on the 28th and 30th of October, 
and also last night (Nov. 2d), and find that for so short an interval the observa- 
tions are well satisfied by a period of revolution of almost exactly four days for 
the outermost, and two days and a half for the closest. They are very faint ob- 
jects ; certainly not half the brightness of the two conspicuous ones ; but all the 
four were last night steadily visible, in the quieter moments of the air, with a 
magnifying power of 118 on the 20 ft. equatorial." 

This discovery would seem to confirm the inference drawn by Mr. Dawes, from 
a discussion of the observations formerly made by Lassell and Struve upon the 
nearest satellite. He considers these observations incompatible with each other. 
•• While Struve's observations indicate a period of three days and twenty hours, 
Lassen's observations indicate a period of only two days and two hours. He 
therefore infers that there must be, at least, two satellites interior to that which 
Herschel denominates the second." He also considers it doubtful whether the 
other satellite discovered by Lassell is really Herschel's third satellite, as stated 
above. 

It would seem, therefore, that at least two, and perhaps three, of the Her- 
schelian satellites have been seen by later observers, and that two new satellite* 
have probably been discovered by Lassell. Accordingly Uranus has certainly 
three satellites, and probably as many as eight. 

Neptune. 

The apparent diameter of Neptune is nearly 3 /A , and its actual diameter is 
41,500 miles. " To two observers it has afforded strong suspicion of being sur- 
rounded with a ring very highly inclined ; and from the observations of Mr. Las- 
sell, M. Otto Struve, and Mr. Bond, it appears to be attended certainly by one, 
and very probably by two satellites, though the existence of the second can hardly 
yet be considered as quite demonstrated." (For the details of the interesting 
history of the discovery of this planet, see Herschel's Outlines of Astronomy, or 
Loomis's Progress of Astronomy.) 

The New Asteroids, 

Astrcea, Hebe, Iris, Flora, Metis, Hygeia, Parthenope, Clio, Egeria, Irene, 
Eunomia. 
Of the dimensions and other physical peculiarities of these planetary bodies,, 
no knowledge has as yet been obtained, further than that they are very small 
bodies, and probably inferior in size to the other four asteroids. They are al) 
of about the ninth apparent magnitude, except Metis, which is of the tenth or 
eleventh. 



NOTE XIII. 

Certain remarkable phenomena were exhibited by Biela's comet at its last 
return (in 1846), an account of which will be found in Note V. 



NOTES XIV, XV. 377 



NOTE XIV. 

The great problem of the determination of the parallax and distance of a 
fixed star, first solved by Bessel, has since been undertaken with success by 
other astronomers. The following is a list of the most reliable determinations 
which have been hitherto obtained : 

a Centauri 0".913 (Henderson). 

61 Cvgni .348 (Bessel). 

a Lyra .261 (Struve). 

Sirius .230 (Henderson). 

- , . A 1n „ Peters, Struve, Preuss, 

Polaris °- 106 i andLindenau. 

In the case of the Pole Star, the estimated error to which the result obtained 
is liable, is l / 9 of the parallax. For the other stars it is a still smaller fraction. 
The parallax of the pole star indicates a distance which light would require 
more than 30 years to traverse. 

The measurements for a Lyrse, as well as for 61 Cygni, were made with a 
micrometer. Professor Henderson determined the parallax of a Centauri, from 
a discussion of a series of observations upon that star made by him, with a large 
mural circle, in the years 1832 and 1833, at the Royal Observatory of the Capo 
of Good Hope. Subsequent observations with a similar but more efficient in- 
strument by Mr. Maclear, have conducted to very nearly the same result. The 
observations by M. Peters were made with the great vertical circle of the Pul- 
kova Observatory. His observations with this instrument upon 61 Cygni gave 
a parallax almost identical with that found by Bessel. This same observer has 
also undertaken to determine the parallax of several other stars, with the fol- 
lowing results: Arcturus (0".127), Iota Ursse Majoris (0".133), 1830 Groom- 
bridge (0".226), Capella (0".046), a Cygni (no measurable parallax). But the 
probable errors are one-half, or more, of the parallaxes found. 



NOTE XV. 

It is an interesting fact, ascertained by M. Argelander, of Bonn, that the 
periods of several of the variable stars are subject to a slow alteration. The 
two stars, Omicron Ceti and Algol, may be cited as examples. It is conjectured 
that these variations of period are periodical. 

Sir John Herschel, in his " Outlines of Astronomy;' gives a list of thirty-four 
variable stars whose periods have been approximately or roughly determined, 
but each year adds to the number. There are many other stars known to be 
variable, but whose periods and limits of variation of brightness are unknown. 

The statement made in the text of the second general fact noticed with 
respect to variable stars should read thus : they pass from their epoch of least 
light to that of their greatest in considerably less time than from their greatest 
to their least. 

" The alterations of brightness in the southern star n Argus, which have been 
recorded, are very singular and surprising. In the time of Halley (1677) it 
appeared as a star of the fourth magnitude. Lacaille, in 1751, observed it of the 
second; in the interval from 1811 to 1815 it was again of the fourth; and 
again, from 1822 to 1826, of the second. On the 1st of February, 1827, it was 
noticed by Mr. Burchell to have increased to the first magnitude, and to equal 
u Crucis. Thence again it receded to the second ; and so continued until the end 
of 1837. All at once, in the beginning of 1838, it suddenly increased in lustre 
so as to surpass all the stars of the first magnitude, except Sirius, Canopus, and 
a Centauri, which last star it nearly equalled. Thence it again diminished, but 
this time not below the first magnitude, until April, 1843, when it had again 
increased so as to surpass Canopus, and nearly equal Sirius in splendor. A 

48 



378 notes xvi-xvm. 

strange field of speculation is opened by this phenomenon. The temporary 
stars heretofore recorded have all become totally extinct. Variable stars, so far 
as they have been carefully attended to, have exhibited periodical alternations, 
in some degree, at least, regular, of splendor and comparative obscurity. But 
here we have a star fitfully variable to an astonishing extent, and whose fluctu- 
ations are spread over centuries, apparently in no settled period, and with no 
regularity of progression. What origin can we ascribe to these sudden flashes 
and relapses ? What conclusions are we to draw as to the comfort or habita- 
bility of a system depending for its supply of light and heat on so uncertain a 
source." (Herschel's Outlines.) 



NOTE XVI. 

It must be conceded that the change in the length of the periods of the vari- 
able stars, noticed in the previous note, is apparently at variance with the 
theory given in the text, since all analogy teaches that the periods of rotation 
should be uniform. Argelander, who has studied the phenomena of variable 
stars more attentively than any other observer, is of the opinion that " the time 
has not come in which we should prepare to frame a theory. The minute 
changes characterizing the phenomena have been too little studied and dis- 
cussed." 



a Andromeda?. 


Orionis 


e Lyrae. 


H Lupi. 


$ Cancri. 


fi Bootis. 



NOTE XVII. 

" Among the most remarkable triple, quadruple, or multiple stars, may be 
enumerated, 

£ Scorpii. 

11 Monocerotis. 

12 Lyncis. 

Of these a Andromedae, n Bootis, and (i Lupi, appear in telescopes even of 
considerable optical power, only as ordinary double stars ; and it is only when 
excellent instruments are used that their smaller companions are subdivided 
and found to be in fact extremely close double stars. £ Lyrae offers the remark- 
able combination of a double -double star. Viewed with a telescope of low 
power, it appears as a close and easily divided double star ; but on increasing 
the magnifying power, each individual is perceived to be beautifully and closely 
double, the one pair being about 2^", the other about Z" asunder. Each of the 
stars, £ Cancri, £ Scorpii, 11 Monocerotis, and 12 Lyncis, consists of a principal 
star, closely double, and a smaller and more distant attendant, while Orionis 
presents the phenomenon of four brilliant principal stars, of the respective 4th, 
6th, 7th, and 8th magnitudes, forming a trapezium, the longest diagonal of which 
is 21 ".4, and accompanied by two excessively minute and very close companions, 
to perceive both of which is one of the severest tests which can be applied to a 
telescope." (Herschel's Outlines.) 



NOTE XVIII. 

Later observations have led to the discovery that the star t Indi has a greater 
proper motion than any other star, — the amount of its annual displacement being 
7".74. 



NOTE XIX. 



379 



An interesting confirmation of the solar motion mentioned in Art. 593 has 
recently been obtained by Mr. Galloway, from a discussion of certain observa- 
tions made at different epochs and by different observers upon eighty-one stars 
of the southern hemisphere. He concludes from bis discussion, that the point 
towards which the sun's motion is directed, is situated in R. A. 260° 1' and H. 
Dec. 34° 23' ; " a result so nearly identical with that afforded by the northern 
hemisphere as to afford a full conviction of its near approach to truth, and what 
may fairly be considered a demonstration of the physical cause assigned." 



NOTE XIX 



The following, according to Herschel, are the places, for 1830, of the principal 
globular clusters, as specimens of their class : — 



E.A. 


N. P. D. 


E. A. 


N. P. D. 


E.A. 


N. P. D. 


h. m. s. 




h. m. s. 




h. m. s. 


o / 


16 25 


163 2 


15 9 56 


87 16 


17 26 51 


143 34 


9 8 33 


154 10 


15 34 56 


127 13 


17 28 42 


93 8 


12 47 41 


159 57 


16 6 55 


112 33 


18 26 4 


114 2 


13 4 30 


70 55 


16 23 2 


102 40 


18 55 49 


150 14 


13 16 38 


136 35 


16 35 37 


53 13 


21 21 43 


78 34 


13 34 10 


60 46 


16 50 24 


119 51 


21 24 40 


91 34 



Many of the nebulous objects in the heavens hitherto classed among resolvable 
nebulae, have lately been resolved by the magnificent reflecting telescope con- 
structed by Lord Rosse ; and many nebula? which have offered no appearance 
of stars to all previous observers, and which were supposed by the elder Her- 
schel to be collections of nebulous matter, have either been partially resolved 
by this telescope, or have assumed in it the appearance of resolv ability. In 
view of these facts it must be conceded, that " although nebula do exist, which 
even in this powerful telescope appear as nebula?, without any sign of resolu- 
tion, it may very reasonably be doubted whether there be really any essential 
physical distinction between nebulae and clusters of stars, at least in the nature 
of the matter of which they consist, and whether the distinction between such 
nebulae as are easily resolved, barely resolvable with excellent telescopes, and 
altogether irresolvable with the best, be any thing else than one of degree, 
arising merely from the excessive minuteness and multitude of the stars, of 
which the latter, as compared with the former, consist." 

Sir James South, who made a trial of Lord Rosse's monster telescope in 
March, 1845, gives the following account of his observations: — "Never before in 
my life did I see such glorious sidereal pictures as this instrument afforded us. 
The most popularly known nebulae observed were the ring nebula in the Canes 
Venatici, which was resolved into stars with a magnifying power of 548, and 
the 94th of Messier, which is in the same constellation, and which was resolved 
into a large globular cluster of stars, not much unlike the well-known cluster in 
Hercules. On subsequent nights observations of other nebulae, amounting to 
some thirty or more, removed most of these from the list of nebulae, where they 
had long figured, to that of clusters ; while some of these latter exhibited a 
sidereal picture in the telescope such as man before had never seen, and which, 
for its magnificence, baffles all description." 

The following are some of the nebulae which have assumed a new and re- 
markable appearance when viewed through Lord Rosse's telescopes, of 3 ft. and 
6 ft. aperture : 

1. The Crab-nebula. To previous observers this curious object presented the 
appearance of an oval resolvable nebula. " Lord Rosse's three feet reflector 
exhibits it with resolvable filaments singularly disposed, springing principally 
from its southern extremity, and not, as is usual, in clusters, irregularly in all 



380 ■ NOTE XIX. 

directions. It is studded with stars, mixed, however, with a nebulosity, probably 
consisting of stars too minute to be recognized." 

2. The Dumb-bell nebula, so named from its resemblance to a dumb-bell, as 
shown by Sir John Herschel's drawing (see Nichol's Architecture of the Heavens), 
in Lord Rosse's 3 ft. telescope, has quite a different appearance, and is seen to 
consist of innumerable stars mixed with nebulosity. 

3. The nebula in the Dog's Ear was formerly described as having the form of 
a ring, divided through about one-third of its course into two separate branches 
or streams, and thus regarded as presenting a singular counterpart to our own 
Milky Way. In Lord Rosse's six feet reflector "the former simple shape is 
transformed into a scroll, apparently unwinding with numerous filaments, and a 
mottled appearance, which looks like the breaking up of a cluster." It has 
accordingly received the designation of the Scroll or Spiral nebula. 

4. The great nebula in Orion has also been divested of the mystery in which 
it has so long remained enshrouded, by the same telescope. Lord Rosse says : 
" I may safely say that there can be little if any doubt as to the resolvability of 
this nebula. We can plainly see that all about the trapezium is a mass of stars ; 
the rest of the nebula also abounding with stars, and exhibiting the characteris- 
tics of resolvability strongly marked." 

Mr. Bond, with the great refractor at Cambridge, has also succeeded in resolv- 
ing the brighter portion of this nebula immediately adjacent to the trapezium, 
or the sextuple star d. 

The great nebula in Andromeda, mentioned in the text, has also been care- 
fully observed with the Cambridge refractor, and decisive evidence obtained of 
its resolvability. 

Detailed descriptions of these two nebulae, as seen with the Cambridge tele- 
scope, accompanied with accurate drawings, have been published by the Messrs. 
Bond (Transactions of the American Academy of Arts and Sciences, vol. hi). 

In the southern hemisphere there are two remarkable nebulous masses of 
light, conspicuously visible to the naked eye, which are known by the name of 
Magellanic Clouds, or Nubeculce (major and minor). Sir John Herschel de- 
scribes them as being in the appearance and brightness of their light not unlike 
portions of the Milky Way of the same apparent size, and round or oval in their 
general form. 

"When examined through powerful telescopes, the constitution of the nube- 
cula?, and especially of the nubecula major, is found to be of astonishing com- 
plexity. The general ground of botli consists of large tracts and patches of 
nebulosity, in every stage of resolution, from light irresolvable with 18 inches of 
reflecting aperture, up to perfectly separated stars like the Milky Way, and 
clustering groups sufficiently insulated and condensed to come under the desig- 
nation of irregular, and in some cases pretty rich clusters. But, besides these, 
there are also nebulae in abundance, both regular and irregular ; globular clus- 
ters in every state of condensation; and objects of a nebulous character quite 
peculiar, and which have no analogue in any other region of the heavens. Such 
is the concentration of these objects, that in the area occupied by the nubecula 
major, not fewer than 2? 8 nebula? and clusters have been enumerated, besides 
50 or 60 outliers, which (considering the general barrenness in such objects of 
the immediate neighborhood) ought certainly to be reckoned as its appendages, 
being about 6} per square degree, which very far exceeds the average of any 
other, even the most crowded parts of the nebulous heavens. In the nubecula 
minor the concentration of such objects is less, though still very striking, 37 hav- 
ing been observed within its area, and 6 adjacent but outlying. The nubecula?, 
then, combine, each within its own area, characters which in the rest of the 
heavens are no less strikingly separated ; viz., those of the galactic and the 
nebular system. Globular clusters (except in one region of small extent) and 
nebulae of regular elliptic forms are comparatively rare in the Milky Way, and 
are found congregated in the greatest abundance in a part of the heavens the 
most remote possible from that circle ; whereas, in the nubecula? they are indis- 
criminately mixed with the general starry ground, and with irregular though 
small nebulae." (Herschel's Outlines of Astronomy.) 



NOTES XX, XXI. 381 



NOTE XX. 

According to Struve, the nebula in Andromeda is l^rlong by 16' broad, and 
thus nearly one-half greater than the moon's disk. Mr. G. P. Bond describes it 
as extending nearly 2^° in length, and upwards of 1° in breadth. 

Since, as stated in Note XIX, nuny'of the nebulae, which were supposed by 
Sir William Herschel to be masses of "nebulous matter, have recently been found 
to consist of stars, it must now be regarded as exceedingly doubtful whether 
any such supposed nebulous masses really exist in space ; and, on the other 
hand, highly probable that all the irresolvable nebulas are only vast beds of 
stars eithertoo remote, or composed of too small or too closely compacted stars, 
to appear otherwise than one general mass of cloudy light in the best tele- 
scopes. 



NOTE XXI 

Struve, of the Pulkova Observatory, in a recent work entitled Etudes d ; As- 
tronomie Stellaire, has undertaken to establish that the stratum of the Milky 
Way is really fathomless (at least in every direction except, perhaps, at right 
angles to the stratum), and shows, by quotations from his later papers on the 
Milky Way. that Sir William Herschel was led finally to entertain the same 
opinion, in opposition to the views he had at first expressed (in 1785). Accord- 
ingly, by comparing the number of stars seen in the field of view of a telescope 
when pointed in two different directions into space, we do not obtain the rela- 
tive distances through to the boundaries of the stratum of the Milky Way, but 
only the relative condensation of the stars, or relative density of the starry 
stratum in the two directions. Every augmentation in the power of the tele- 
scope brings into view, in these directions, other stars before invisible. 

Struve remarks : " It may be asked why astronomers have generally main- 
tained the old theory concerning the Milky Way, propounded in 1785, although 
it had been entirely abandoned by the author himself, as we have demonstrated. 
I believe that the explanation must be sought in two circumstances. It was a 
complete system, imposing from the boldness and geometric precision of its 
construction, and which the author has never revoked as a whole. In his trea- 
tises, published since 1802, we meet with only partial views, but which are suf- 
ficient, when they are compared together, to exhibit the final idea of the great 
astronomer." 

Sir John Herschel does not give his assent to the opinion expressed by Struve. 
He remarks : — " Throughout by far the larger portion of the extent of the Milky 
Way in both hemispheres, the general blackness of the ground of the heavens 
on which its stars are projected, and the absence of that innumerable multitude 
and excessive crowding of the smallest visible magnitudes, and of glare pro- 
duced by the aggregate light of multitudes too small to affect the eye singly, 
which the contrary supposition would seem to necessitate, must, we think, be 
considered unequivocal indications that its dimensions in directions where these 
conditions obtain, are not only not infinite, but that the space-penetrating power 
of our telescopes suffices fairly to pierce through and beyond it." 

If it be true that the stratum of the Milky Way is really fathomless — that 
infinite space is occupied by an infinite number of shining stars, the central suns 
of planetary systems clustered around the'm, as first suggested by Kant, then it 
has been shown by Qlbers that the aspect of the sky should be that of a vault 
9hining in all directions with a lustre similar to that of the sun. The conclusion, 
therefore, is inevitable, either that the bed of stars in which our sun is posited 
is not infinite in extent, or that space is not perfectly transparent ; in other 
words, that the light coming from the stars suffers a partial extinction, propor- 
tional in amount to the distance traversed by it. The latter view was advo- 
cated by Olbers, and is also adopted by Struve, who by means of this concep- 
tion endeavors to reconcile his views of the boundless extent of our firmament 



382 note xxn. 

with the feeble luminosity of the sky. He conceives, upon a detailed investiga- 
tion, that the actual luminosity of the sky in different directions is adequately 
explained, in accordance with his theory of the unlimited extent of the stratum 
of the Milky Way, if it be allowed that the light of the stars suffers an extinc- 
tion of only ' 100 in traversing a distance equal to that of a star of the first mag- 
nitude. Upon this supposition the extinction for the most distant stars visible 
in telescopes would amount to 8S per cent. 

Herschel urges, in opposition to this theory, that 1; if applicable to any. it is 
equally so to every part of the galaxy. We are not at liberty to argue that at 
one part of its circumference our view is limited by this sort of cosmical veil 
which extinguishes the smaller magnitudes, cuts off the nebulous light of distant 
masses, and closes our view in impenetrable darkness ; while at another we are 
compelled by the clearest evidence telescopes can afford to believe that star- 
strewn vistas lie open, exhausting their powers and stretching out beyond their 
utmost reach, as is proved by that very phenomenon which the existence of 
such a veil would render impossible, viz., infinite increase of number and dimi- 
nution of magnitude, terminating in complete irresolvable nebulosity." 



NOTE XXII. 

Or rather, when the planets are compared with respect to density, it will be 
seen that they may be divided into two classes, viz. : one class, comprising Mer- 
cury, Venus, the Earth, and Mars, each of which has a density nearly equal to 
unity ; and a second class, consisting of Jupiter, Saturn, Uranus, and Xeptune, 
whose density is between 0.13 and 0.23. 

It is a curious fact that the same classification holds with respect to magni- 
tude and period of rotation. 



< 



4 



# 



> 



A 



« 



TABLE I. 1 

Latitudes and Longitudes from the Meridian of Greenwich, of 
some cities, and other conspicuous places. 



Names of Places. 


Latitude. 


Longitude 
in Degrees. 


Longitude 
in Time. 


Albany, Capitol, 


New York, 


o / // 
42 39 3 N 


o / // 
73 44 49 W 


h m s 

4 54 59.3 


Altona, Obs., 


Denmark, 


53 32 45 N 


9 56 39 E 


39 46.6 


Baltimore, Bait. Mon% 


Maryland, 


39 17 23 N 


76 37 30 W 


5 6 30 


Berlin, Obs., 


Germany, 


52 31 13 N 


13 23 52 E 


53 35.5 


Boston, State House, 


Massach'ts, 


42 21 23 N 


71 4 9W 


4 44 16.6 


Bremen, 06,-?., 


Germany, 


53 4-36N 


8 48 58 E 


35 15.9 


Brunswick, Bowdoin Coll., 


Maine, 


43 53 N 


69 55 1 W 


4 39 40 


Canton, 


China, 


23 8 9N 


113 16 54 E 


7 33 8 


Cape of Good Hope, Obs., 


Africa, 


33 56 3 S 


18 28 45 E 


1 13 55.0 


Cape Horn, 


S. America, 


55 58 41 S 


67 10 53 W 


4 23 43 


Charleston, »S1 Mich's Ch. 


S. Carolina, 


32 46 33 N 


79 57 27 W 


5 19 49.8 


Charlottesville, Univers., 


Virginia, 


38 2 3N 


78 31 29 W 


5 14 6 


Cincinnati, Fort Wash , 


Ohio, 


39 5 54 N 


84 27 W 


5 37 48 


Copenhagen, Obs., 


Denmark, 


55 40 53 N 


12 34 57 E 


50 19.8 


Dorpat, Obs., 


Russia, 


53 22 47 N 


26 43 45 E 


1 46 55 


Dublin, Obs., 


Ireland, 


53 23 13 N 


6 20 30 W 


25 22 


Edinburgh, Obs., 


Scotland, 


55 57 23 N 


3 10 54 W 


12 43.6 


Gotha, 06s., 


Germany, 


50 56 5 N 


10 44 6 E 


42 56.4 


Gottingen, Obs., 
Greenwich, Obs., 


Germany, 


51 31 48 N 


9 56 37 E 


39 46.5 


England, 


51 28 39 N 








Konigsberg, 06s., 
London, St. Paul's Ch., 


Prussia, 


54 42 50 N 


20 30 7 E 


1 22 05 


England, 


51 30 49 N 


5 48 W 


23 


Marseilles, 06s., 


France, 


43 17 50 N 


5 22 15 E 


21 29.0 


Milan, 06s., 


Italy, 


45 28 IN 


9 11 48 E 


36 47.2 


Naples, 06s., 


Italy, 


40 51 47 N 


14 15 4 E 


57 3 


New Haven, College, 


Connecticut, 


41 18 30 N 


72 56 45 W 


4 51 47 


New Orleans, City Hall, 
New York, City Hall, 


Louisiana, 


29 57 45 N 


90 6 49 W 


6 27 


New York, 


40 42 40 N 


74 1 8W 


4 58 4.5 


Palermo, 06s., 


Italy, 


38 6 44 N 


13 21 24 E 


53 25.6 


Paramatta, 06s., 


New Holl'd, 


33 48 50 S 


151 1 34 E 


10 4 6.3 


Paris, 06s., 


France, 


48 50 13 N 


2 20 24 E 


9 21.6 


Petersburgh, 06s., 
Philadelphia, Ind'ce Hall, 


Russia, 


59 56 31 N 


30 18 57 E 


2 1 15 .8 


Pennsylv'a, 


39 56 59 N 


75 9 54 W 


5 39.6 


Point Venus, 


Otaheite, 


17 29 21 S 


149 28 55 W 


9 57 56 


Princeton, College, 


New Jersey, 


40 20 41 N 


74 39 33 W 


4 58 38.2 


Providence, University, 


Rhode Isl'd, 


41 49 22 N 


71 24 48 W 


4 45 39 2 


Quebec, Castle, 


L. Canada, 


46 49 12 N 


71 16 W 


4 45 4 


Richmond, Capitol, 


Virginia, 


37 32 17 N 


77 27 28 W 


5 9 50 


Rome, Roman College, 


Italy, 


41 53 52 N 


12 28 40 E 


49 54.7 


Savannah, Exchange, 


Georgia, 


32 4 56 N 


81 8 18 W 


5 24 33 


Schenectady, 


New York, 


42 48 N 


73 55 W 


4 55 40 


Stockholm, 06s., 


Sweden, 


59 20 31 N 


18 3 44 E 


1 12 15 


Turin, 06s., 


Italy,. 


45 4 6N 


7 42 6E 


30 48.4 


Vienna, 06s., 


Austria, 


48 12 35 N 


16 23 E 


1 5 32 


Wardhus, 


Lapland, 


70 22 36 N 


31 7 54 E 


2 4 32 


Washington, Capitol, 


Dist. Colum. 


38 53 34 N 


77 1 80 W 


5 8 6 



2 TABLE II. Elements of the Planetary Orbits. 

Epoch for Vesta, Juno, Ceres, and Pallas, July 23d, 1831, mean noon at 
Berlin : for the other planets, Jan. 1, 1801, mean noon at Greenwich.* 



Planet's 
Name. 



Mercury 

Venus 

Earth 

Mars 

Vesta 

Juno 

Ceres 

Pallas 

Jupiter 

Saturn 

Uranus 



Inclination to 
the Ecliptic. 



7 9.1 
3 23 28 5 

1 51 6.2 

7 7 57.3 

13 2 10.0 

10 36 55.7 

34 35 49.1 

1 18 51.3 

2 29 35.7 
46 28.4 



Sec. Var. 



+ 18.2 

— 46 

— 0.2 

— 12 

— 44 

— 22.6 

— 15.5 

+ 3.1 



Longitude of As- 
cending Node. 



45 57 30.9 
74 51 55 

48 3.5 

103 20 28.0 
170 52 34.5 

80 53 49.7 
172 38 29.8 

98 26 189 
111 56 37.4 

72 59 35.3 



Sec. Var. 



70.44 
51.10 

41.67 
26 



+ 25 

+ 57.18 
4- 51.12 
-f- 23.58 



Longitude of 
Perihelion. 



74 21 469 
128 43 53 1 

99 31 9.9 
332 23 56.6 
249 11 37. 

54 17 12.7 
147 41 23.5 
121 5 05 

11 8 34.6 

89 9 29.8 
167 31 16.1 



Secj Var. 



-h 



93.22 

78.30 
103.15 
109.71 
157 



+ 202 



i 



94 59 
115.68 

87.44 



pi„„ o t> ATo^a Mean Distance from Mean Distance from Eccentricity in Parts 
nanet s i\ ame. Snn nr s om i„ v i«. s,m in mmp« nfthoSomi.oTK 



Sun. or Semi-axis. 



Sun in Miles. 



of the Semi-axis. 



Sec. Variation. 



Mercury 

Venus 

Earth 

Mars 

Vesta 

Juno 

Ceres 

Pallas 

Jupiter 

Saturn 

Uranus 



3870981 
0.7233316 
1.0000000 
1.5236923 
2.3614800 
2.6694600 
27709100 
2.7726300 
5.2027760 
9.5387861 
19.1823900 



36814000 

68787000 

95103000 

144908000 

224584000 

253874000 

263522000 

263685000 

494797000 

907162000 

1824290000 



020551494 
000686074 
01678357 
0.09330700 
0.03856000 
0.25556000 
0.07673780 
0.24199800 
004816210 
0.05615050 
0.04661080 



+ .000003866 

— .000062711 

— .000041630 
4- .000090176 
-j- .000004009 

— .000005830 

+ .000159350 

— .000312402 

— .000025072 



Planet's Name 



Mean Longitude at 
the Epoch. 



Mean Sidereal Period in Motion in mean Lon. 
Mean Solar Days. in 1 yr. of 365 days. 



Mean Daily Motion 
in Longitude, 



Mercury 

Venus 

Earth 

Mars 

Vesta 

Juno 

Ceres 

Pallas 

Jupiter 

Saturn 

Uranus 



166 43.6 

11 33 3 

100 39 133 

64 22 555 

84 47 3 2 

74 39 43 6 

307 3 25.6 

290 38 11.8 

112 15 23.0 

135 20 6.5 

177 48 23.0 



d 

87. 

224. 

365. 

636 

1325 

1593 

1684. 

1636. 

4332, 

10759. 



9692580 
7007869 
2563770 
9796458 
4850000 
0670000 
7350000 
3050000 
5848212 
2198174 
8208296 



53 43 3 6 
224 47 29.7 
—0 14 19.5 
191 17 9.1 



30 20 31.9 

12 13 36.1 

4 17 45.1 



/ // 

4 5 32.6 

1 36 7.8 
59 8.3 
31 26.7 
16 17.9 
13 33.7 
12 49.4 
12 48.7 
4 59.3 
2 06 
42.4 



TABLE III.— Elements of Moon's Orbit. Epoch, Jan. 1, 1801. 



Mean inclination of orbit 
Mean longitude of node at epoch 
Mean longitude of perigee at epoch - 
Mean longitude of moon at epoch 
Mean distance from earth, or semi-axis 
Eccentricity in parts of semi-axis 



Mean sidereal revolution - 

Mean tropical do. 

Mean synodical do. - 

Mean anomalistic do. - 

Mean nodical do. - 

Mean revolution of nodes; sider. 

Mean revolution of perigee ; sider. 



d k m s 
27 7 43 11.5 

27 7 43 47 ; 
29 12 44 2.9 
27 13 18 37.4 
27 5 5 36.0 

6793d. 279; trop. 

: 3232d.57534; trop.: 



5 8 47.9 

13 53 17.7 

266 10 7.5 

118 17 8.3 

59r. 964350 

0.0548442 

d 

27.32166142 
2732153242 
29.5305S872 
27.55459950 
27.21222222 
6793d. 17707 
3231d.4751 



Elements of Neptune.— Mem distance, 30.0368000; Period, (i0120 d .7100000; Eccentricity, 
0087195 : Inclination of orbit, 1° 46' 59".() ; Long, of Node, 130° 5' 1 1".0 ; Long, of Perihelion, 
470 12' 5&".7; M.Long, at Epoch, 330° 44' 41".8 ; Epoch, 1848, Jan. 1, Oh. G. T. 

* For an accurate table of the Elements of all the Asteroids, see Note III. 



TABLE IV, 3 

Diameters, Volumes, Masses, &c, of Sun, Moon, and Planets. 





Apparent Diameter, 


Equatorial 
Diameter.* 


Equatorial 
Diameter, 
in Miles.* 


Volume. 


Least. 


At Mean 
Distance. 


Greatest. 


Mercury 

Venus 

Earth 

Mars 
: Jupiter 
, Saturn 

Uranus 
i Neptune 

i 

Sun 
Meon 


5.0 
9.6 

3.6 
30.0 

/ // 

31 31.0 
29 21.9 


6.5 
16.5 

5.8 

36.9 

16.2 

3.9 

3.0 

1 11 

32 1,8 

31 7.0 


12.0 
61.2 

18.3 
45.9 

t n 

32 35,6 

33 31.1 


0.396 
0.984 
1.000 
0.517 
10.976 
9.987 
4.353 
5.236 

112.020 
0.273 


3140 

7800 
7926 
4100 
87000 
79160 
34500 
41500 

887870 
2163 


0.062 

0.952 

1.000 

0.138 

1233.412 

900.000 

82.759 

144.008 

1410366.376 
0,020 





Mass.f 


Density.}: 


Gravity. 


Sidereal Rotation.} 


Light and 
Heat. 










h. m. s. 




Mercury 


i 

4lFB"57"5T 


1.12 


0.47 


24 5 28.3 


6.680 


• Venus 


1 

4013T9 


0,92 


0.93 


23 21 21,9 


1.911 


Earth 


1 


1.00 - 


1,00 


33 56 4.1 


1.000 


3 5 9 5 5 1 


Mars 


1 

2BT0337 


0.95 


0.50 


24 37 20.4 


,431 


Jupiter 


1 

104 7 "STT 


0,24 


2.85 


9 55 26.6 


.037 


Saturn 


1 
3J0 j-FcTo 


0,14 


1.03 


10 29 16.8 


,011 


Uranus 


I 
24 9 01 


0.24 


0.76 




,003 


Neptune 


1 


0.14 


0.69 




.001 


, Sun 


1 


G.25 


28.65 


607 48 




Moon 


l 

3 1?43406 


0.57 


0.15 


27 7 43 





TABLE V, 
Elements of the Retrograde Motion of the Planets. 



Planets. 


Arc of 


Duration of 


Elongation at the 


Synodic 


Retrogradation. 


Retrogradation. 


Stations. 


Revolution. 




o o 


d h d h 


o ' O " 


dayc 


Mercury 


9 22 tc 15 44 


23 12 to 21 12 


14 49 to 20 51 


116 


Venus 


14 35 to 17 12 


40 21 to 43 12 


27 40 to 29 41 


584 


Mars 


10 6 to 19 35 


60 18 to 80 15 


128 44 to 146 37 


780 


Jupiter 


9 51 to 9 59 


116 18 to 122 12 


113 35 to 116 42 


399 


Saturn 


6 41 to 6 55 


138 18 to 135 9 


107 25 to 110 46 


378 


Uranus 


3 36 


151 


103 30 


370 



Satellites of Neptune. — " One only has certainly been observed — its approximate 
period being 5d. 20h. 50m. 45s.; distance about 12 radii of the planet." 

* According to Herschel, except the diameters of the Sun and Moon. 

t According to Encke, with the exception of the mass of Neptune, which is Professor Peirce'a 
determination from Bond's and Lassel's observations of the satellite. By Leverrier'a second 
■determination the mass of Mercury is 3^0 o"o^o • 

% According to Hansen and Madler, in the case of the planets. 



TABLE VL 



Elements of the Orbits of the Satellites. 

The distances are expressed in equatorial radii of the primaries, 
periods are expressed in mean solar days. 

L Satellites of Jupiter. 



The 



Sat. 



Mean Distance. 



6.04853 

9.62347 

15.35024 

26.99835 



Sidereal 
Revolution. 



1 18 27 33.506 

3 13 14 36.393 

7 3 42 33.362 

16 16 31 49.702 



Inclination of 

Orbit to that 

of Jupiter. 



o r " 
3 5 30 

Variable. 
Variable. 
2 58 48 



Epoch Mass - y that of 
of Ele-' Jupiter being 
ments. 1,000,000,000. 



Jan. 1, 
1801. 



17328 
23235 

88497 
42659 



II. Satellites of Saturn. 



Name and 
Order of 
Satellite. 


Mean 
Distance. 


Sidereal 
Revolution. 


M. Long, at the 
Epoch. 


Eccentricity 

and Perisatur- 

nium. 


Epoch 
of Ele- 
ments. 








O ' 


» 






1. Mimas 


3.3607 


22 37 22.9 


256 58 


48 




1790.0 


2. Enceladus 


4.3125 


1 8 53 6.7 


67 41 


36 




1836.0 


3. Tethys 


5.3396 


1 21 18 25.7 313 43 


48 


0.04(0— 54 ( ? ) 


Ditto 


4. Dione 


6.8398 


2 17 41 8.9 


327 40 


48 


0.02(?)— 42(?) 


Ditto 


5. Rhea 


9.5528 


4 12 25 10.8 


353 44 





0.02(?)— 95(?) 


Ditto 


6. Titan 


22.1450 


15 22 41 25.2 


137 21 


24 


). 029314 
f 256° 38' 


1830.0 


7. Hyperion 


28.± 


22 12 ? 








8. Iapetus 


64.3590 


79 7 53 40.4 


269 37 


48 




1790.0 



The longitudes are reckoned in the plane of the ring from its descending node with 
the ecliptic. The first seven satellites move in or very nearly in its plane fthat of the 
8th lies about half-way between the planes of the ring and of the planet's orbit. The 
apsides of Titan have a direct motion of 30' 28" per annum in longitude (on the ecliptic), 

III. Satellites of Uranus. 



Sat. 


Mean 
Distance. 


Sidereal 
Revolution. 


Epochs of Passage 

through Ascending 

Node of Orbits. G. T. 


Inclination to Ecliptic. 


1 
2 
3 
4 
5 
6 


17.0 
19.8 (?) 
22.8 
45.5 (?) 
91.0 


d h m s 

4(?) 

8 16 56 31.3 
10 23(?) 
13 11 7 12.6 
38 2 (?) 
107 12(?) 


1787, Feb. 16th, 10 
1787, Jan. 7th, 28 


The orbits are inclined at 
an angle of about 78° 58' 
to the ecliptic in a plane 
whose ascending node is 
in long. 165° 30' (Equinox 
of 1798). Their motion 
is retrograde. The orbits 
are nearly circular. 



TABLE VII. Saturn's Ring. 



Exterior diameter of exterior ring 




... 176,418 miles. 
... 155 272 " 


Exterior diameter of interior ring 




... 151,690 " 
... 117,339 " 


Equatorial diameter of the body 

Interval between the planet and interic 

Interval of the rings 

Thickness of the rings not exceeding ... 


>r ring 


... 79,160 " 
... 19,090 " 




... 1,791 " 
250 " 


Ditto, according to Professor Bond, not 


exceeding.... 


50 " 



VIII. 



Mean Astronomical Refractions. 
Barometer 30 in. Thermometer, Fah. 50°. 



Ap.Alt. 


Refr. 


Ap. Alt 


Refr. 


Ap. Alt. 


Refr. 


1 Alt. 

42° 


Refr. 


0° 0' 


33' 51" 


4° 0' 


11' 52" 


12° 0' 


4' 28.1" 


r.4.6" 


5 


32 53 


10 


11 30 


10 


4 24.4 


43 


1 2.4 


10 


31 58 


20 


11 10 


20 


4 20.8 


44 


1 0.3 


15 


31 5 


30 


10 50 


30 


4 17.3 


45 


0.58.1 


20 


30 13 


40 


10 32 


40 


4 13.9 


46 


56.1 


25 


29 24 


50 


10 15 


50 


4 10.7 


47 


54.2 


30 


28 37 


S 5 


9 58 


13 


4 7-5 


48 


52.3 


35 


27 51 


10 


9 42 


10 


4 4.4 


49 


50.5 


40 


27 6 


20 


9 27 


20 


4 1.4 


50 


48.8 


45 


26 24 


30 


9 11 


30 


3 58.4 


51 


47.1 


50 


25 43 


40 


8 58 


40 


3 55.5 


52 


45.4 


55 


25 3 


50 


8 45 


50 


3 52.6 


53 


43.8 


1 


24 25 


6 


8 32 


14 


3 49.9 


54 


42.2 


5 


23 48 


10 


8 20 


10 


3 47.1 


55 


40.8 


10 


23 13 


20 


8 9 


20 


3 44.4 


56 


39.3 


15 


22 40 


30 


7 58 


30 


3 41.8 


57 


37.8 


20 


22 8 


40 


7 47 


40 


3 39.2 


58 


36.4 


25 


21 37 


50 


7 37 


50 


3 36.7 


59 


35.0 


30 


21 7 


7 


7 27 


15 


3 34.3 


60 


33.6 


35 


20 38 


10 


7 17 


'15 30 


3 27.3 


61 


32.3 


40 


20 10 


20 


7 8 


16 


3 20.6 


62 


31.0 


45 


19 43 


30 


6 59 


16 30 


3 14.4 


63 


29.7 


50 


19 17 


40 


6 51 


17 


3 8.5 


64 


28.4 


55 


18 52 


50 


6 43 


1 17 30 


3 2.9 


65 


27.2 


2 


IS 29 


8 


6 35 


| 18 


2 57.6 


66 


25.9 


5 


18 5 


10 


6 28 


| 19 


2 47.7 


67 


24.7 


10 


17 43 


20 


6 21 


20 


2 38.7 


68 


23.5 


15 


17 21 


30 


6 14 


21 


2 30.5 


69 


22.4 


20 


17 


40 


6 7 


' 22 


2 23.2 


70 


21.2 


25 


16 40 


50 


6 


23 


2 16.5 


71 


19.9 


30 


16 21 


9 


5 54 


24 


2 10.1 


72 


18.8 


35 


16 2 


10 


5 47 


25 


2 4.2 


73 


17.7 


40 


15 43 


20 


5 41 


26 


1 58.8 


74 


16.6 


45 


15 25 


30 


5 36 


27 


1 53.8 


75 


15.5 


50 


15 8 


40 


5 30 


28 


1 49.1 


76 


14.4 


55 


14 51 


50 


5 25 


29 


1 44.7 


77 


13.4 


3 


14 35 


10 


5 20 


30 


1 40.5 


78 


12.3 


5 


14 19 


10 


5 15 


31 


1 36.6 


79 


11.2 


10 


14 4 


20 


5 10 


32 


1 33.0 


80 


10.2 


15 


13 50 


30 


5 5 


33 


1 29.5 


81 


9.2 


20 


13 35 


40 


5 1/ 


34 


1 26.1 


82 


8.2 


25 


13 21 


50 


4 56 


35 


1 23.0 


83 


7.1 


30 


13 7 


11 


4 51 


36 


1 20.0 


84 


6.1 


35 


12 53 


10 


4 47 


37 


1 17.1 


85 


5.1 


40 


12 41 


20 


4 43 


38 


1 14.4 


86 


4.1 


45 


12 28 


30 


4 39 


39 


1 11.8 


87 


3.1 


50 


12 16 i 


40 


4 35 


40 


1 9.3 


88 


2.0 


55 


12 3 1 


50 


4 31 


41 


1 6.9 


89 


1.0 



TABLE IX 



Corrections of Mean Refractions. 



Ap.Alt. 


diffoi 
+1B 


dif. far 
— 1°F. 


1 Ap.Alt. 


Dif. for 
-KB. 


Dif. for 
— 1°F. 


Ap. Alt. 


Dif. f 01 

-j-lB. 


(Dif. for 
— ]°F. 


Alt. 


Dif. for 
+1B. 


Dif. fori 
- -1° F. 


O ' 


// 


rf 


O ' 


*r 


" 


o /• 


tr 


" 


o 


" 


" 


a o 


74 


8.1 


4 


24.1 


1.70 


12 


9.00 


0.556 


42 


2.16 


0.130 


5 


71' 


7.6 


10 


23.4 


1.64 


10 


8.86 


.548 


43 


2.09 


.125 


10 


69 


7.3 


20 


22.7 


1.58 


20 


8.74 


.541 


44 


2.02 


.120 


15 


67 


7.0 


30 


22.0 


1.53 


30 


8.63 


.533 


45 


1.95 


.116 


20 


65 


6.7 


40 


21.3 


1.48 


40 


8.51 


.524 


46 


1.88 


.112 


25 


63 


6.4 


50 


20.7 


1.43 


50 


8.41 


.517 


47 


1.81 


.108 


30 


61 


6.1 


5 


20.1 


1.38 


13 


8.30 


.509 


48 


1.75 


.104 


35 


59 


5.9 


10 


19.6 


1.34 


10 


8.20 


.503 


49 


1.69 


.101 


40 


58 


5.6 


20 


19.1 


1.30 


20 


8.10 


.496 


50 


1.63 


.097 


45 


56 


5.4 


30 


18.6 


1.26 


30 


8.00 


.490 


51 


1.58 


.094 


50 


55 


5.1 


40 


18.1 


1.22 


40 


7.89 


.482 


52 


1.52 


.090 


55 


53 


4.9 


50 


17.6 


1.19 


50 


7.79 


.476 


53 


1.47 


.088 


1 


52 


4.7 


6 


17.2 


1.15 


14 


7.70 


.469 


54 


1.41 


.085 


5 


50 


4.6 


10 


16.8 


1.11 


10 


7.61 


.464 


55 


1.36 


.082 


10 


49 


4.5 


20 


16.4 


1.09 


20 


7.52 


.458 


56 


1.31 


.079 


15 


48 


4,4 


30 


16.0 


1.06 


30 


7.43 


.453 


57 


1.26 


.076 


20 


46 


4.2 


40 


15.7 


1.03 


40 


7.34 


.448 


58 


1.22 


.073 


25 


45 


4,0 


50 


15.3 


1.00 


50 


7.26 


.444 


59 


1.17 


.070 


30 


44 


3.9 


7 


150 


0.98 


15 


7.18 


.439 


60 


1.12 


.067 


35 


43 


3.8 


10 


14.6 


.95 


15 30 


6.95 


.424 


61 


1.08 


.065 


40 


42 


3.6 


20 


14.3 


.93 


16 


6.73 


.411 


62 


1.04 


.062 


45 


40 


3.5 


30 


14.1 


.91 


16 30 


6.51 


.399 


63 


*.99 


.060 


50 


39 


3.4 


40 


13.8 


.89 


17 


6.31 


.386 


64 


.95 


.057 


55 


39 


3.3 


50 


13.5 


.87 


17 30 


6.12 


.374 


65 


.91 


.055 


2 


38 


3.2 


8 


13.3 


.85 


18 


5.94 


.362 


66 


.87 


.052 


5 


37 


3.1 


10 


13.1 


.83 


19 


5.61 


.340 


67 


.83 


.050 


10 


36 


3.0 


20 


12.8 


.82 


20 


5.31 


.322 


68 


.79 


.047 


15 


36 


2.9 


30 


12.6 


.80 


21 


5.04 


.305 


69 


.75 


.045 


20 


35 


2.8 


40 


12.3 


.79 


22 


4.79 


.290 


70 


.71 


.043 


25 


34 


2.8 


50 


12.1 


.77 


23 


4.57 


.276 


71 


.67 


.040 


30 


33 


2.7 


9 


11.9 


.76 


24 


4.35 


.264 


72 


.63 


.038 


35 


33 


2.7 


10 


11.7 


.74 


25 


4.16 


.252 


73 


.59 


.036 


40 


32 


2.6 


20 


11.5 


.73 


26 


3.97 


.241 


74 


.56 


033 


45 


32 


2.5 


30 


11.3 


.72 


27 


3.81 


.230 


75 


.52 


.031 


50 


31 


2.4 


40 


11.1 


.71 


28 


3.65 


.219 


76 


.48 


.029 


55 


30 


2.3 


50 


11.0 


.70 


29 


3.50 


.209 


77 


45 


.027 


3 


30 


2.3 


10 


10.8 


.69 


30 


3.36 


.201 


78 


.41 


.025 


5 


29 


2.2 


10 


10.6 


.67 


31 


3.23 


.193 


79 


.38 


.023 


10 


29 


2.2 


20 


10.4 


.65 


32 


3.11 


.186 


80 


.34 


.021 


15 


28 


2.1 


30 


10.2 


.64 


33 


2.99 


.179 


81 


.31 


.018 


20 


28 


2.1 


40 


10.1 


.63 


34 


2.88 


.173 


82 


.27 


.016 


25 


27 


2.0 


50 


9.9 


.62 


35 


2.78 


.167 


83 


.24 


.014 


30 


27 


2.0 


11 


9.8 


.60 


36 


2.68 


.161 


84 


.20 


.012 


35 


26 


2.0 


10 


9.6 


.59 


37 


2.58 


.155 


85 


.17 


.010 


40 


26 


1.9 


20 


9.5 


.58 


38 


2.49 


.149 


86 


.14 


.008 


45 


25 


1.9 


30 


9.4 


.57 


39 


2.40 


.144 


87 


.10 


.006 


50 


25 


1.9 


40 


9.2 


.56 


40 


2.32 


.139 


88 


.07 


.004 


55 


25 


1.8 1 


50 I 


9.1 


.55 


41 


2.24 


.134 


89 


.03 


.002 



TABLE X. 



Parallax of the Sun, on the first day of each Month : the mean 
horizontal Parallax being assumed = 8 ",60. 



Alti- 


Jan. 


Feb. 


March. 


April. 


May. 


June. 


July. 


tude. 


Dec. 


Nov. 


Oct. 


Sept. 


Aug. 


o 


// 


// 


// 


„ 


// 


/■/ 


" 





8.75 


8.73 


8.67 


8.60 


8.53 


8.48 


8.46 


5 


8.73 


8.69 


8.64 


8.56 


8.50 


8.44 


8.42 


10 


8.62 


8.59 


8.54 


8.47 


8.40 


8.35 


8.33 


15 


8.45 


8.43 


8.38 


8.30 


8.24 


8.19 


8.17 ; 


20 


8.22 


8.20 


8.15 


8.08 


8.01 


7.97 


7.95 


25 


7.93 


7.91 


7.86 


7.79 


7.73 


7.68 


7.67 


30 


7.58 


7.56 


7.51 


7.45 


7.39 


7.34 


7.33 


35 


7.17 


7.15 


7.11 


7.04 


6.99 


6.94 


6.93 


40 


6.70 


6.68 


6.64 


6.59 


6.53 


6.49 


6.48 


45 


6.19 


6.17 


6.13 


6.08 


6.03 


5.99 


5.98 


50 


5.62 


5.61 


5.58 


5.53 


5.48 


5.45 


5.44 


55 


5.02 


5.01 


4.98 


4.93 


4.89 


4.86 


4.85 


60 


4.37 


4.36 


4.34 


4.30 


4.26 


4.24 


4.23 


65 


3.70 


3.69 


3.67 


3.63 


3.60 


3.58 


3.57 


70 


2.99 


2.98 


2.97 


2.94 


2.92 


2.90 


2.S9 


75 


2.26 


2.26 


2.25 


2.23 


2.21 


2.19 


2.19 


80 


1.52 


1.52 


1.51 


1.49 


1.48 


1.47 


1.47 


85 


0.76 


0.76 


0.76 


0.75 


0.74 


0.74 


0.74 ] 


90 


0.00 


0.00 


0.00 


0.00 


000 


0.00 


0.00 



TABLE XL 

Semi-diurnal Arcs. 



Lat. 



Declination. 



10 : 



i-y- 



20° 



25o 



30o 



6 4 

6 5 

6 6 

6 7 

6 9 



6 12 

6 14 

17 

20 
24 
29 

35 



6 43 



6 4 
6 7 
6 11 
6 15 
6 19 
6 23 
6 28 
6 34 
6 41 
6 49 

6 58 

7 11 
7 29 



5 
II 
16 
22 

29 
36 
43 
52 
2 
14 
30 
51 
8 20 



7 

15 
22 
30 
39 
49 
59 
11 
25 
43 
5 

8 36 

9 25 ll2 



12 

24 

36 

49 

2 

18 

35 

56 

21 

S 54 

9 42 

12 



TABLE XII. 



Equation of Time, ,to convert Apparent Time into Mean Time 
Argument, Mean Longitude of the Sun. 





0* 


Is 


Us 


Ills 


IVs 


V* 





min. sec. 


min. sec. 


min. sec. 


min. sec. 


min. sec. 


min. sec. 





+ 6 58.4 


— 1 29.7 


— 3 38.7 


+ 1 27.0 


+ 6 4.1 


+ 2 49.7 


1 


6 39.7 


1 42.0 


3 34.2 


140.1 


6 6.3 


2 34.5 


2 


6 20.9 


1 53.8 


3 29.1 


1 53.1 


6 8.0 


2 18.9 


3 


6 2.1 


2 5.2 


3 23.5 


2 6.0 


6 9.1 


2 2.8 


4 


5 43.3 


2 15.9 


3 17.3 


2 18.9 


6 9.5 


1 46.4 


5 


5 24.5 


2 26.1 


3 10.7 


2 31.7 


6 9.3 


1 29.5 


6 


5 5.7 


2 35.9 


3 3.5 


2 44.3 


6 8.5 


1 12.3 


7 


4 46.9 


2 45.0 


2 56.0 


2 56.7 


6 7.2 


54.6 


8 


4 28.2 


2 53.6 


2 47.9 


3 8.9 


6 5.2 


36.6 


9 


4 9.6 


3 1.8 


2 39.5 


3 20.8 


6 2.5 


+ 18.2 


10 


3 51.1 


3 9.3 


2 30.5 


3 32.5 


5 59.3 


— 0.4 


11 


3 32.6 


3 16.3 


2 21.2 


3 43.9 


5 55.4 


19.5 


12 


3 14.3 


3 22.8 


2 11.5 


3 55.0 


5 51.0 


38.8 


13 


2 56.2 


3 28.6 


2 1.4 


4 5.8 


5 45.8 


58.4 


14 


2 38.3 


3 33.9 


1 51.0 


4 16.3 


5 40.1 


1 18.2 


15 


2 20.5 


3 38.6 


140.1 


4 26.5 


5 33.7 


1 38.3 


16 


2 3.0 


3 42.7 


1 29.0 


4 36.3 


5 26.7 


1 58.5 


17 


1 45.7 


3 46.3 


1 17.6 


4 45.7 


5 19.2 


2 19.1 


18 


1 28.6 


3 49.2 


1 5.9 


4 54.7 


5 11.1 


2 39.8 


19 


1 11.7 


3 51.5 


54.1 


5 3.3 


5 2.3 


3 0.7 


20 


55.2 


3 53.3 


42.0 


5 11.3 


4 53.0 


3 21.6 


21 


39.1 


3 54.4 


29.6 


5 18.9 


4 43.1 


3 42.8 


22 


23.3 


3 55.0 


17.1 


5 26.0 


4 32.7 


4 4.0 


23 


+ 7.8 


3 55.0 


— 4.4 


5 32.6 


4 21.6 


4 25.3 


24 


— 7.3 


3 54.5 


+ 8.4 


5 3S.6 


4 10.1 


4 46.6 


25 


22.0 


3 53.3 


21.5 


5 44.2 


3 57.9 


5 8.1 


26 


36.3 


3 51.5 


34.5 


5 49.3 


3 45.3 


5 29.5 


27 


50.3 


3 49.2 


47.6 


5 53.9 


3 32.1 


5 51.0 


28 


1 3.S 


3 46.2 


1 0.7 


5 57.8 


3 IS. 5 


6 12.3 


29 


1 16.9 


3 42.8 


1 13.8 


6 1.2 


3 4.3 


6 33.7 


{ 30 


— 1 29.7 


— 3 38.7 


+ 1 27.0 


+ 6 4.1 


+ 2 49.7 


— 6 54.9 

-A 



TABLE XIII. 

Secular Variation of Equation of Time. 
Argument, Sun's Mean Longitude. 





O 


Is 


lis 


Ills 


IVs 


Vs 


sec. 


sec. 


sec. 


sec. 


sec. 


sec. 


sec.\ 





— 3 


+ 4 


+ 11 


+ 14 


f 13 


+ 9 1 


3 


2 


5 


11 


14 


13 


8 i 


6 


1 


6 


12 


14 


12 


3 | 


9 . 


— 1 


6 


12 


15 


12 


7 j 


12 





7 


12 


14 


12 


7 i 


15 


+ 1 


8 


13 


14 


11 


6 


18 


2 


8 


13 


14 


11 


6 


21 


2 


9 


14 


14 


10 


5 


24 


3 


9 


14 


14 


10 


5 


27 


4 


10 


14 


14 


9 


4 


30 


+ 4 


+ 11 


+ 14 


T 13 


+ 9 


+ 4 



TABLE XII 



9 



Equation of Time, to convert Apparent Time into Mean Time. 
Argument, Mean Longitude of the Sun. 





VI* 


VII« 


VIII* 


IX* 


X* 


XI* 


o 


min. sec. 


min. sec. 


min. sec. 


min. sec. 


min. sec. 


min. sec. 





— 6 54.9 


— 15 18.9 


— 13 58.7 


— 1 30.6 


+ 11 30.0 


+ 14 3.1 


1 


7 16.1 


15 27.9 


13 43.0 


1 0.2 


11 47.0 


13 56.0 


2 


7 37.2 


15 36.1 


13 26.3 


— 29.8 


12 3.3 


13 48.4 


3 


7 58.3 


15 43.7 


13 8.9 


+ 0.6 


12 18.7 


13 40.1 


4 


8 19.1 


15 50.5 


12 50.5 


31.0 


12 33.4 


13 31.1 


5 


8 39.8 


15 56.5 


12 31.4 


1 1.3 


12 47.2 


13 21.6 


6 


9 0.2 


16 1.8 


12 11.6 


131.4 


13 0.1 


13 11.4 


7 


9 20.5 


16 6.3 


1151.1 


2 1.3 


13 12.2 


13 0.7 


8 


9 40.6 


16 9.9 


1129.9 


2 31.0 


13 23.5 


12 49.4 


9 


10 0.3 


16 12.9 


11 7.9 


3 0.5 


13 33.9 


12 37.4 


10 


10 19.8 


16 15.1 


10 45.4 


3 29.7 


13 43.6 


12 25.0 


11 


10 38.9 


16 16.5 


10 22.0 


3 58.6 


13 52.3 


12 12.2 


12 


10 57.8 


16 17.0 


9 58.1 


4 27.1 


14 0.2 


11 58.9 


13 


11 16.2 


16 16.6 


9 33.5 


4 55.2 


14 7.3 


11 45.1 


14 


11 34.4 


16 15.4 


9 8.4 


5 22.9 


14 13.5 


11 30.9 


15 


11 52.1 


16 13.4 


8 42.6 


5 50.2 


14 18.9 


11 16.3 


16 


12 9.5 


16 10.4 


8 16.4 


6 17.1 


14 23.4 


11 1.1 


17 


12 26.5 


16 6.7 


7 49.6 


6 43.5 


14 27.2 


10 45.6 


18 


12 42.9 


16 2.1 


7 22.5 


7 9.3 


14 30.0 


10 29.7 


19 


12 58.9 


15 56.6 


6 54.9 


7 34.6 


14 32. 1 


10 13.5 


20 


13 14.4 


15 50.1 


6 27.0 


7 59.3 


14 33.3 


9 56.9 


21 


13 29.5 


15 42.9 


5 58.5 


8 23.4 


14 33.7 


9 40.1 


22 


13 44.1 


15 34.8 


5 29.7 


8 46.9 


14 33.3 


9 23.0 


23 


13 58.0 


15 25.8 


5 0.5 


9 9.8 


14 32.2 


9 5.7 J 


24 


14 11.4 


15 16.0 


4 31.0 


9 32.0 


14 30.2 


8 48.0 i 


25 


14 24.1 


15 5.2 


4 1.4 


9 53.5 


14 27.5 


8 30.2 ! 


26 


14 36.3 


14 53.6 


3 31.6 


10 14.3 


14 24.0 


8 12.2 ! 


27 


14 47.9 


14 41.1 


3 1.5 


10 34.4 


14 19.9 


7 54.0 ! 


28 


14 58.8 


14 27.7 


2 31.3 


10 53.8 


14 15.0 


7 35.5 


29 


15 9.2 


14 13.6 


2 1.0 


11 12.3 


* 14 9.4 


7 17.0 : 


30 


— 15 18.9 


— 13 58.7 


— 1 30.6 


+ 11 30.0 


+ 14 3.1 


+ 6 58.4 

j 



TABLE XIII. 

Secular Variation of Equation of Time, 
Argument, Sun's Mean Longitude. 





VI* 


VII* 


VIII* 


IX* 


X* 


XI* 


o 


sec. 


sec. 


sec. 


sec. 


sec. 


sec. 





+ 4 


— 2 


—10 


—15 


—15 


—10 


3 


3 


3 


10 


15 


14 


10 


6 


3 


4 


11 


15 


14 


9 


o 


2 


4 


12 


15 


14 


8 


12 


1 


5 


12 


15 


13 


8 


15 


+ 1 


6 


13 


15 


13 


7 


18 





7 


13 


15 


12 


6 


21 





7 


14 


15 


12 


5 


24 


—1 


8 


14 


15 


11 


5 


27 


2 


9 


15 


15 


11 


4 


30 


— 2 


—10 


—15 


—15 


—10 


-3 



10 



TABLE XIV. 
Perturbations of Equation of Time. 
III. 



II. 




sec. 


100 

sec. 


200 


300 


400 

sec. 


50C 

sec. 


600 


700 
sec. 


800 


900 1 


1000 




sec. 


sec. 


sec. 


sec. 


sec. 


sec. 





1.4 


0.8 


1.0 


1.7 


1.7 


1.2 


0.7 


0.4 


0.6 


1.4 


1.4 


100 


1.2 


1.4 


1.1 


1.0 


1.6 


1.8 


1.1 


0.7 


0.6 


0.7 


1.2 


200 


0.9 


1.0 


1.2 


1.2 


1.2 


1.5 


1.7 


1.1 


0.5 


0.7 


0.9 


300 


0.7 


1.1 


1.1 


0.9 


1.2 


1.4 


1.5 


1.6 


1.2 


0.5 


0.7 


400 


0.5 


0.6 


1.2 


1.2 


0.8 


1.0 


1.6 


1.7 


1.5 


1.2 


0.5 


500 


1.0 


0.5 


0.6 


1.2 


1.4 


0.8 


0.8 


1.5 


1.9 


1.5 


1.0 


600 


1.7 


1.0 


0.4 


0.5 


1.2 


1.4 


0.9 


0.6 


1.3 


2.0 


1.7 


700 


1.9 


1.8 


1.1 


0.4 


0.4 


1.1 


1.6 


1.1 


0.7 


1.2 


1.9 


800 


1.2 


1.8 


1.8 


1.2 


0.4 


0.3 


1.0 


1.6 


1.2 


0.7 


1.2 


900 


0.7 


1.1 


1.7 


1.8 


1.2 


0.6 


0.2 


0.8 


1.6 


1.3 


0.7 


1000 


1.4 


0.8 


1.0 


1.7 


1.7 


1.2 


0.7 


0.4 


0.6 


1.4 


1.4 


.,.! 


IV. 








sec. 


sec. 


sec. 


sec. 


sec. 


sec. 


sec. 


sec. 


sec. 


sec. 


sec. 





0.6 


0.7 


0.5 


0.3 


0.2 


0.6 


0.7 


0.5 


0.2 


0.1 


0.6 


100 


0.2 


0.7 


0.6 


0.5 


0.2 


0.3 


0.6 


0.9 


0.5 


0.2 


0.2 


200 


0.2 


0.4 


0.6 


0.5 


0.4 


0.3 


0.4 


0.6 


0.5 


0.5 


0.2 


300 


0.4 


0.2 


0.5 


0.5 


0.5 


0.4 


0.4 


0.4 


0.5 


0.5 


0.4 


400 


0.5 


0.4 


0.4 


0.4 


0.4 


0.4 


0.5 


0.5 


0.4 


0.4 


0.5 


500 


0.4 


0.5 


0.5 


0.5 


0.4 


0.4 


0.3 


0.4 


0.5 


0.3 


0.4 


600 


0.3 


0.3 


0.5 


0.6 


0.4 


0.4 


0.3 


0.5 


0.7 


0.4 


0.3 


700 


0.4 


0.2 


0.3 


0.6 


0.6 


0.4 


0.2 


0.2 


0.7 


0.7 


0.4 


800 


0.6 


0.3 


0.2 


0.3 


0.7 


0.6 


0.3 


02 


0.3 


0.8 


0.6 


900 


0.8 


0.5 


0.3 


0.1 


0.4 


0.7 


0.5 


0.3 


0.1 


0.5 


0.8 


1000 


0.6 


0.7 


0.5 


0.3 


0.2 


0.6 


0.7 


0.5 


0.2 


0.1 


0.6 


II. 


V. 
* 




sec. 


sec. 


sec. 


sec. 


sec. 


sec. 


sec. 


sec. 


sec. 


sec. 


sec. 





1.0 


1.0 


1.1 


1.2 


1.1 


1.0 


0.7 


0.4 


0.6 


0.9 


1.0 


100 


0.9 


0.9 


0.8 


1.0 


1.3 


1.3 


1.0 


0.7 


0.4 


0.5 


0.9 


200 


0.5 


0.7 


0.7 


0.8 


1.0 


1.0 


1.1 


1.2 


0.9 


0.3 


0.5 


300 


0.2 


0.5 


0.7 


0.7 


0.8 


1.2 


1.5 


1.5 


1.1 


0.5 


0.2 


400 


0.3 


0.2 


0.5 


0.7 


0.7 


0.9 


1.3 


1.4 


1.4 


1.0 


0.3 


500 


0.8 


0.3 


0.2 


0.5 


0.7 


0.7 


1.0 


1.4 


1.4 


1.4 


0.8 


600 


1.3 


0.7 


0.3 


0.3 


0.5 


0.7 


0.9 


1.1 


1.4 


1.6 


1.3 


700 


1.5 


1.1 


0.7 


0.3 


0.4 


0.5 


0.8 


1.0 


1.2 


1.4 


1.5 


800 


1.3 


1.3 


1.0 


0.7 


0.4 


0.4 


0.6 


0.8 


1.0 


1.2 


1.3 


900 


1.1 


1.2 


1.2 


1.0 


0.8 


0.6 


0.5 


0.6 


0.9 


1.1 


1.1 


1000 


1.0 


1.0 


1.1 


1.2 


1.1 


1.0 


0.7 


0.4 


0.6 


0.9 


1.0 


Moon and Nutation. 




sec. 


sec. \ sec. B sec. 


1 sec. 


sec. 


sec. 


sec. 


sec. 


sec. 


sec. 


I. 


0.5 


0.8 1 1.0 1.0 


08 


0.5 


0.2 


0.0 


0.0 


0.2 


0.5 


N. 


0.1 


0.1 | 0.2 ! 0.2 


1 0.2 


0.2 


0.2 


0.2 


0.2 


0.1 


0.1 



Constant 3*.0 



TABLE XV. 



U 



"br converting any given day into the decimal part of a year 
of 365 days. 



Day 


Jan. 


Feb. 


March 


April 


May 


June 


1 


.000 


.085 


.162 


.247 


.329 


.414 


2 


.003 


.088 


.164 


.249 


.331 


.416 


3 


.006 


.090 


.167 


.252 


.334 


.419 


4 


.008 


.093 


.170 


.255 


.337 


.422 


5 


.011 


.096 


.173 


.258 


.340 


.425 


6 


.014 


.099 


.175 


.260 


.342 


.427 


7 


.016 


.101 


.178 


.263 


.345 


.430 


8 


.019 


.104 


.181 


.266 


.348 


.433 


9 


.022 


.107 


.184 


.268 


.351 


.436 


10 


.025 


.110 


.186 


.271 


.353 


.438 


11 


.027 


.112 


.189 


274 


.356 


.441 


12 


.030 


.115 


.192 


.277 


.359 


.444 


13 


.033 


.118 


.195 


.279 


.362 


.446 


14 


.036 


.121 


.197 


.282 


.364 


,449 


15 


.038 


.123 


.200 


.285 


.367 


.452 


16 


.041 


.126 


.203 


.288 


.370 


.455 


17 


.044 


.129 


.205 


.290 


.373 


.458 


18 


•046 


.132 


.208 


.293 


.375 


.460 


19 


.049 


.134 


.211 


.296 


.378 


.463 


20 


.052 


.137 


.214 


.299 


.381 


.466 


21 


.055 


.140 


.216 


.301 


.384 


.468 


22 


.058 


.142 


.219 


.304 


.386 


.471 


23 


.060 


,145 


.222 


.30? 


.389 


.474 


24 


.063 


.148 


.225 


.310 


.392 


.477 


25 


.066 


.151 


.227 


.312 


.395 


.479 


26 


.068 


.153 


.230 


.315 


.397 


.482 


27 


.071 


.156 


.233 


.318 


.400 


.485 


28 


.074 


.159 


.236 


.321 


.403 


.488 


29 


.077 




.238 


.323 


.405 


.490 


30 


.079 




.241 


.326 


.408 


.493 


3L 


.082 




.241 




.411 


J 



12 



TABLE XV.. Continued. 



For converting any given day into the decimal part of a year 
of 365 days. 



Day 


July 


August 


Sept. 


Oct 


Nov. 


Dec. 


1 


.496 


.581 


.666 


.748 


.833 


.915 


2 


.499 


.584 


.668 


.751 


.836 


.918 


3 


.501 


.586 


.671 


.753 


.838 


.921 


4 


.504 


.589 


.674 


.756 


.841 


.923 


5 


.507 


.592 


.677 


.759 


.844 


.926 


6 


.510 


.595 


.679 


.762 


846 


.929 


7 


.512 


.597 


.682 


.764 


.849 


.931 


8 


.515 


.600 


.685 


.767 


.852 


.934 


9 


.518 


.603 


.688 


.770 


855 


.937 


10 


.521 


.605 


.690 


.773 


.858 


.940 


11 


.523 


.608 


.693 


.775 


.860 


.942 


12 


526 


611 


.696 


.778 


.863 


.945 


13 


.529 


614 


.699 


.781 


.866 


.948 


14 


.532 


.616 


.701 


.784 


.868 


.951 


15 


.534 


.619 


.704 


.786 


.871 


.953 


16 


.537 


.622 


.707 


.789 


.874 


.956 


17 


.540 


.625 


.710 


.792 


877 


.959 


18 


.542 


.627 


.712 


.795 


879 


.962 


19 


.545 


.630 


.715 


.797 


.882 


.964 


20 


.548 


.633 


.718 


.800 


885 


.967 


21 


.551 


.636 


.721 


.803 


888 


.970 


22 


.553 


.638 


.723 


805 


890 


.973 


23 


.556 


.641 


.726 


.808 


893 


.975 


24 


.559 


.644 


.729 


.811 


896 


.978 


25 


.582 


.647 


.731 


.814 


.899 


.981 


26 


.564 


.649 


.734 


.816 


.901 


.984 


27 


.567 


.652 


.737 


.819 


.904 


.986 


28 


.570 


.655 


.740 


.822 


.907 


.989 


29 


.573 


.658 


.742 


.825 


.910 


.992 


30 


.575 


.660 


.745 


.827 


.912 


.995 


31 


.578 


.663 




.830 




.997 



TABLE XVX 



13 



For converting time into decimal parts of a day. 



Hours 


Minutes 


Second^ 


h. 




m. 


m. 




s. 


s. 




1 


.04167 


1 


.00069 


31 


.02153 


1 


.00001 


• 31 


.00036 


2 


.08333 


2 


.00139 


32 


.02222 


2 


.00002 


32 


.00037 


3 


.12500 


3 


.0020S 


33 


.02292 


3 


.00003 


v , 33 


.0003S 


4 


.16667 


4 


.00278 


34 


.02361 


4 


.00005 


34 


.00039 


5 


.20833 


5 


.00347 


35 


.02430 


5 


.00006 


35 

36 
37 


.00040 


6 


.25000 


6 


.00417 


36 
37 


.02500 


6 


.00007 


.00042 


7 


.29167 


7 


.00486 


.02569 


7 


.00008 


.00043 


8 


.33333 


8 


.00556 


38 


.02639 


8 


.00009 


■ 38 


.00044 


9 


.37500 


9 


.00625 


39 


.02708 


9 


.00010 


39 


.00045 


10 


.41667 


10 


.00694 


40 


.02778 


10 


.00012 


; 40 

:• 41 


.00046 


11 


.45833 


11 


.00764 


41 


.02847 


• 11 


.00013 


.00047 


12 


.50000 


12 


.00333 


42 


.02917 


12 


.00014 


1 42 

j 43 


.00049 


13 


.54167 


13 


.00903 


43 


.02986 


13 


.00015 


.00050 


14 


.58333 


14 


.00972 


44 


.03056 


14 


.00016 


8 44 


.00051 


15 


.62500 


15 


01042 


45 


.03125 


15 


.00017 


[45 


.00052 


16 


.66667 


16 


.01111 


46 


.03194 


16 


.00018 


) 46 


.00053 


17 


.70833 


17 


.01180 


47 


.03264 


17 


.00020 


;47 
i 48 


.00054 


18 


.75000 


18 


.01250 


48 


.03333 


18 


.00021 


.00056 


19 


.79167 


19 


.01319 


49 


.03403 


19 


.00022 


i 49 


.00057 


20 


.83333 


20 


.013S9 


50 


.03472 


20 


.00023 


j 50 


.00058 


21 


.87500 


21 


01458 


51 


.03542 


21 


.00024 


51 


.00059 


22 


.91667 


22 


.01528 


52 


.03611 


22 


.00025 


52 


.00060 


23 


.95833 


23 


01597 


53 


.03680 j 


23 


.00027 


53 

54 


.00061 


24 


1.00000 


24 


.01667 


54 


.03750 1 


24 


.00028 


.00062 






25 


.01736 


55 


.03819 ! 


25 


.00029 
.00030 


55 


.00064 


i 




26 


.01805 


56 


.03S89 j 


26 


56 


.00065 






27 


.01875 


57 


.03958 ; 


27 


.00031 


57 


.00066 






28 


.01944 


58 


.04028 


28 


.00032 


58 


.00067 






29 


.02014 


59 


.04097 i 


29 


.00034 


59 


.00068 






30 


.020S3 


60 


.04167 | 


30 1 


.00035 I 


60 j 


.00069 



14 



TABLE XVII. 



For converting Minutes and Seconds of a degree, into the 
decimal division of the same. 



Minutes 




Seconds 


"i 


1 


.01667 31 


.51667 


1 


.00028 31 


.00861 


2 


.03333 


32 


.53333 


2 


.00056 


32 


.00889 


o 


.05000 


33 


.55000 


3 


.00083 


33 


.00917 


4 


.06667 
.08333 


34 


.56667 


4 


.00111 


34 


.00944 


5 


35 


.58333 


5 


.00139 


35 


.00972 


6 


.10000 


36 


.60000 


6 


.00167 


36 


.01000 


7 


.11667 


37 


.61667 


7 


.00194 


37 


.01028 


8 


.13333 


38 


.63333 


8 


.00222 


38 


.01056 


9 


.15000 


39 


.65000 


9 


.00250 


39 


.01083 


10 


.16667 


40 


.66667 


10 


.00278 


40 


.01111 


11 


.18333 


41 


.68333 


11 


.00306 


41 


.01139 


12 


.20000 


42 


.70000 


12 


.00333 


42 


.01167 


13 


.21667 


43 


.71667 


13 


.00361 


43 


.01194 


14 


.23333 


44 


.73333 


14 


.00389 


44 


.01222 


15 


.25000 


45 


.75000 


15 


.00417 


45 


.01250 


16 


.26667 


46 


.76667 


16 


.00444 


46 


.01278 


17 


.28333 


47 


.78333 


17 


.00472 


47 


.01306 


18 


.30000 


48 


.80000 


IS 


.00500 


48 


.01333 


19 


.31667 


49 


.81667 


19 


.00528 


49 


.01361 


20 


.33333 


50 


.83333 


20 


.00556 


50 


.01389 


21 


.35000 


51 


.85000 


21 


.00583 


51 


.01417 


22 


.36667 


52 


.86667 


22 


.00611 


52 


.01444 


23 


.38333 


53 


.88333 


23 


.00639 


53 


.01472 


24 


.40000 


54 


.90000 


24 


.00667 


54 


.01500 


25 


.41667 


55 


.91667 


25 


.00694 


55 


.01528 


26 


.43333 


56 


.93333 


26 


.00722 


56 


.01556 


27 


.45000 


57 


.95000 


27 


.00750 


57 


.01583 


28 


.46667 


58 


.96667 


28 


.00778 


58 


.01611 


29 


.48333 


59 


.98333 


29 


.00806 


59 


.01639 


30 


.50000 


60 


1.00000 


30 


.00833 


60 


.01667 







TABLE XVIII. 










15 


Sun's Epochs. 


Years. ! M. Long. 


Long.Peri. I 


I 


" 1 


III 


IV 


v 1 


N 


VI 

989 


VII ! 


1830 j 


3 0'" 

9 10 37 46.9 


3 ° ' " 

9 10 54 


22S 


279 


169 


598 


758 


519 


362 i 


1831 


9 10 23 27.4 


9 10 1 55 1 


58S 


278 


793 


130 


842 


573 


235 


396 i 


1832B. 


9 10 9 7.9 


9 10 2 57 


948 


27S 


418 


661 


926 


627 


482 


430 | 


1833 


9 10 53 56.8 


9 10 3 59 


342 


280 ! 


47 


194 


11 


681 


764 


464 


1834 


9 10 39 37.3 


9 10 5 


702 


279 


671 


725 


95 


734 


11 


498 


1835 


9 10 25 17.S 


9 10 6 2 


62 


279 | 


296 


256 


179 


788 


257 


532 


1836B. 


9 10 10 58.4 


9 10 7 3 


422 


278 


920 


788 


264 


842 


504 


566 


1837 


9 10 55 47.2 


9 10 8 5 


816 


280 


549 


321 


348 


895 


787 


600 


1838 


9 10 41 27.8 


9 10 9 6 


176 


279 


173 


852 


432 


949 


33 


634 


1839 


9 10 27 8.3 


9 10 10 8 


536 


279 


798 


383 


517 


3 


279 


668 


1840B. 


9 10 12 48.8 


9 10 11 9 


896 


278 


422 


915 


601 


56 


526 


702 


1841 


9 10 57 37.7! 


9 10 12 11 


290 


280 


51 


447 


685 


110 


809 


736 


1842 


9 10 43 IS. 2 


9 10 13 12 


650 


279 


676 


979 


770 


164 


55 


770 


1843 


9 10 28 5S.8 


9 10 14 14 


10 


279 


300 


510 


854 


218 


301 


804 


1844B. 


9 10 14 39.3 


9 10 15 15 


370 


278 


924 


41 


938 


272 


548 


838 | 


IS45 


9 10 59 28.2 


9 10 16 17 


764 


280 


553 


574 


23 


325 


831 


872 | 


1846 


9 10 45 8.7 


9 10 17 19 


124 


280 


177 


106 


107 


379 


77 


906 


1847 


9 10 30 49.2 


9 10 18 20 


484 


279 


802 


637 


191 


433 


324 


940 


I1848B. 


9 10 16 29.8 


9 10 19 22 


844 


278 


427 


168 


276 


487 


570 


974 


1849 


9 11 1 18.6 


9 10 20 23 


23S 


280 


55 


700 


360 


540 


853 


8 


1850 


9 10 46 59.2 


9 10 21 25 


598 


280 


680 


231 


444 


5«4 


99 


41 


1851 


9 10 32 39.7 


9 10 22 26 


958 


279 


304 


762 


529 


648 


346 


75 


1852B. 


9 10 18 20.2 


9 10 23 28 


319 


278 


929 


294 


613 


701 


592 


109 


1853 


9 11 3 9.1 


9 10 24 29 


713 


2S0 


557 


827 


697 


755 


875 


143 


1854 


9 10 48 49.6 


9 10 25 31 


73 


280 


182 


358 


782 


809 


121 


177 


1855 


9 10 34 30.2 


9 10 26 32 


433 


279 


806 


889 


866 


863 


368 


211 


'1856B 


9 10 20 10.7 


9 10 27 34 


793 


279 


430 


421 


950 


916 


614 


245 


.1857 


9 11 4 59.6 


9 10 28 35 


187 


281 


60 


953 


35 


970 


897 


279 


1858 


9 10 50 40.1 


9 10 29 37 


547 


280 


684 


485 


119 


24 


144 


313 


1859 


9 10 36 20.7 


9 10 30 39 


907 


279 


308 


16 


203 


78 


390 


347 


:1860B 


9 10 22 1.2 


9 10 31 40 


267 


279 


933 


547 


288 


131 


636 


381 


1861 


9 11 6 50.1 


9 10 32 42 


661 


281 


562 


80 


372 


185 


919 


415 


1862 


9 10 52 30.6 


9 10 33 43 


21 


280 


186 


612 


456 


239 


166 


449 


1863 


9 10 38 11.1 


9 10 34 45 


381 


280 


810 


143 


541 


292 


412 


483 


I1864B 


9 10 23 51.7 


9 10 35 46 


741 


279 


435 


674 


625 


346 


659 


517 


1865 


9 11 8 40.5 


9 10 36 48 


135 


281 


64 


207 


709 


400 


941 


551 


1866 


9 10 54 21.1 


9 10 37 49 


495 


280 


688 


738 


794 


453 


188 


585 


1867 


9 10 40 1.6 


9 10 38 51 


855 


280 


313 


270 


878 


507 


434 


619 


1868B 


. 9 10 25 42.2 


9 10 39 52 


215 


279 


937 


801 


962 


561 


681 


653 


1869 


9 11 10 31.0 


9 10 40 54 


609 


281 


566 


334 


47 


615 


963 


687 


1870 

i — 


9 10 56 11.6 


9 10 41 56 


969 


280 


190 


865 131 


668 


210 


721 



16 



TABLE XIX. 

Surfs Motions for Months. 




M. Long. 1 Per, 



0.0 

1 33 18.2 
1 29 3 11.4 

1 2* 8 19.8 

2 28 42 29.7 
2 29 41 3S.0 



3 28 16 39.6 20 



3 29 15 47.9 

4 28 49 57.9 

4 29 49 6.2 

5 28 24 7.8 

5 29 23 16.1 

6 28 57 26.1 

6 29 56 34.4 

7 29 30 44.2 

8 29 52.6 
S 29 4 54.1 

9 4 2.5 



9 29 38 12.5 
10 37 20.7 

10 29 12 22.3 | 56 

11 11 30.6! 56 



I 




II 



III 



IV 





47 


S5 


138 


45 


993 


162 


263 


86 


27 


164 


267 


87 


42 


246 


401 


131 


76 


249 


405 


132 


59 


329 


534 


175 


92 


331 


53S 


176 


110 


414 


672 


220 


144 


416 


676 


221 


129 


496 


S06 


263 


163 


499 


S10 


265 


182 


580 


943 


309 


216 


583 


948 


310 


233 


665 


81 


354 


268 


668 


86 


355 


250 


748 


215 


397 


284 


750 


219 


399 


300 


832 


353 


443 


333 


835 


357 


444 


313 


915 


486 


4S6 


347 


917 


491 


488 



IN 



VI 





125 
141 
178 
266 
302 

355 
391 
480 
516 
569 
605 

694 20 

730 ! 20 
819 I 23 
S55 23 
908 25 
944 25 

33 28 

69 2S 

121 31 



VII 



49 158 3] 



TABLE XX. 



Surfs Motions for Days and Hours. 



Days 


M. Long. 


Per. 


I 


II 


ni 


i V _ 


V 


N 


VI 


VII 


Hrs. 


Long. 


I 
VI 


II 


III 





O ' " 


tr 








1 




- 








/ ,/ 








j 1 


0.0 





























1 


2 27.8 


1 








2 


59 8.3 





34 


3 


4 


1 








36 





2 


4 55.7 


3 








3 


1 58 16.7 





68 


5 


9 


3 








73 





3 


7 23.5 


4 







; 4 


2 57 25.0 





101 


8 


13 


4 


1 





109 





4 


9 51.4 


6 







: 5 


3 56 33.3 


1 


135 


11 


18 


6 


1 


1 


145 





5 


12 19.2 


7 


1 




! 6 


4 55 41.6 


1 


169 


14 


22 


7 


1 




181 





6 


14 47.1 


8 


1 




7 


5 54 50.0 


1 


203 


16 


27 


9 


1 




218 


1 


7 


17 14.9 


10 


1 




! 8 


6 53 58.3 


1 


236 


19 


31 


10 


2 




254 


1 


8 


19 42.8 


11 


1 




• 9 


7 53 6.6 


1 


270 


22 


36 


12 


2 




290 


1 


9 


22 10.6 


13 


1 


2 


! 10 


8 52 15.0 


1 


304 


25 


40 


13 


2 




327 


1 


10 


24 38.5 


14 


1 


2 


i 11 


9 51 23.3 


2 


338 


27 


44 


15 


2 


1 


363 


1 


11 


27 6.3 


16 


1 


2 


1 12 


10 50 31.6 


2 


371 


30 


49 


16 


2 


2 


399 


1 


12 


29 34.2 


17 


1 


2 


1 13 


11 49 40.0 


2 


405 


33 


53 


17 


3 


2 


435 


1 


13 


32 2.0 


18 


1 


2 


14 


12" 48 48.3 


2 


439 


36 


58 


19 


3 


2 


472 


1 


14 


34 29.9 


20 


2 


3 


15 


13 47 56.6 


2 


473 


38 


62 


20 


3 


2 


508 


2 


15 


36 57.7 


21 


2 


3 


16 


14 47 4.9 


2 


506 


41 


67 


22 


3 


2 


544 


o 


16 


39 25.6 


23 


2 


3 


17 


15 46 13.3 


3 


540 


44 


71 


23 


4 


2 


581 


2 


17 


41 53.4 


24 


2 


3 


18 


16 45 21.6 


3 


574 


47 


76 


25 


4 


2 


617 


2 


18 


44 212 


25 


2 


3 


19 


17 44 29.9 


3 


608 


49 


80 


26 


4 


3 


653 


2 


19 


46 49.1 


27 


2 


4 


20 


18 43 38.3 


3 


641 


52 


85 


28 


4 


3 


690 


2 


20 


49 16.9 


28 


2 


4 


21 


19 42 46.6 


3 


675 


55 


89 


29 


5 


3 


726 


2 


21 


51 44.S 


30 


2 


4 


i 22 


20 41 54.9 


4 


709 


58 


93 


31 


5 


3 


762 


2 


22 


54 12.6 


31 


2 


4 


■ 23 


21 41 3.3 


4 


743 


60 


98 


32 


5 


3 


798 


2 


23 


56 40.5 


32 


3 


4 


24 


22 40 11.6 


4 


777 


63 


102 


33 


5 


o 


835 


2 


24 


59 8.3 


34 


3 


4 


1 25 


23 39 19.9 


4 


810 


66 


107 


35 


5 


4 


871 


2 












1 86 


24 38 28.2 


4 


844 


68 


111 


36 


6 


4 


907 


2 












27 


25 37 36.6 


4 


878 


71 


116 


38 


6 


4 


943 


2 












28 


26 36 44.9 


5 


912 


74 


120 


39 


6 


4 


9S0 


2 










. 


29 


27 35 53.2 


5 


945 


77 


125 


41 


6 


4 


16 


3 












30 


28 35 1.6 


5 


979 


79 


129 


42 


7 


4 


52 


3 












31 


29 34 9.9 


5 


13 


82 


134 


44 


7 


4 


89 


1 3 













TABLE XXI. TABLE XXII. 17 

Sun's Motions for Minutes and Seconds. e< *. ^ !? . . ^ °J 

J the JiiCiiptic. 



Min. 


Long. 


1 
Min. 


Long. 


Sec. 

1 


Lon. 
0.0 


1 Sec. 

I 
31 


Lon. 
1.3 


1 


2.5 


31 


1 16.4 


2 


4.9 


32 


1 18.8 


2 


0.1 


32 


1.3 


3 


7.4 


33 


1 21.3 


3 


0.1 


! 33 


1.4 


4 


9.9 


34 


1 23.8 


4 


0.2 


1 34 


1.4 


5 


12.3 


35 


1 26.2 


5 


0.2 


j 35 


1.4 


6 


14.8 


36 


1 28.7 


6 


0.2 


36 


1.5 


7 


17.2 


37 


1 31.2 


7 


0.3 


37 


1.5 


8 


19.7 


38 


1 33.6 


8 


0.3 


38 


1.6 


9 


22.2 


39 


1 36.1 


9 


0.4 


39 


1.6 


10 


24.6 


40 


1 38.6 


10 


0.4 


40 


1.6 


11 


27.1 


41 


1 41.0 


11 


0.5 


'41 


1.7 


12 


29.6 


42 


1 43.5 


12 


0.5 


42 


1.7 


13 


32.0 


43 


1 46.0 


13 


0.5 


43 


1.8 


14 


34.5 


44 


1 48.4 


14 


0.6 


44 


1.8 


15 


37.0 


45 


1 50.9 


15 


0.6 


45 


1.8 


16 


39.4 


46 


1 53.3 


16 


0.7 


46 


1.9 


17 


41.9 


47 


1 55.8 


17 


0.7 


47 


1.9 


18 


44.4 


48 


1 58.3 


18 


0.7 


48 


2.0 


19 


46.8 


49 


2 0.7 


19 


0.8 


49 


2.0 


20 


49.3 


50 


2 3.2 


20 


0.8 


50 


2.0 


' 21 


51.7 


51 


2 5.7 


21 


0.9 


51 


2.1 


22 


54.2 


52 


2 8.1 


22 


0.9 


52 


2.1 


23 


56.7 


53 


2 10.6 


23 


0.9 


53 


2.2 


24 


59.1 


54 


2 13.1 


24 


1.0 


54 


2.2 


25 


1 1.6 


55 


2 15.5 


25 


1.0 


55 


2.3 


26 


1 4.1 


56 


2 18.0 


26 


1.1 


56 


2.3 


27 


1 6.5 


57 


2 20.5 


27 


1.1 


57 


2.3 


28 


1 9.0 


58 


2 22.9 


28 


1.1 


58 


2.4 


29 


1 11.5 


59 


2 25.4 


29 


1.2 


59 


2.4 


30 


1 13.9 


60 


2 27.8 


30 


1.2 


60 


2.5 



i ° ' 


Years! 23 27 


1835 


38 80 


1836 


38.35 


1837 


37.89 


1838 


37.43 


1839 


36.98 


1840 


36.52 


1841 


36.06 


1842 


35.61 


1843 


35.15 


1844 


34.69 


1845 


34.23 


1846 


33.78 


1S47 


33.32 


184S 


32.86 


1849 


32.41 


1850 


31.95 


1851 


31.49 


1852 


31.04 


1S53 


30.58 


1854 


30.12 


1S55 


29.66 


1856 


29.21 


1857 


28 75 


1858 


28 29 


1859 


2784 


1860 


27.38 


1861 


26 92 


1862 


26.47 


1863 


26.01 


1864 


25.55 



TABLE XXIII. 

Sun's Hourly Motion. 
Argument. Sun's Mean Anomaly. 





O* 


I* 


II* 


Ills 


TVs 


Ys 


o 
30 
20 
10 




o 


10 
20 
30 


2 32.92 
2 32.84 
2 32.59 
2 32.20 


2 32.20 
2 31.67 
2 31.02 
2 30.28 


2 30.23 
2 29.46 
2 28.61 
2 27.74 


2 27.74 
2 26.89 
2 26.07 
2 25.32 


2 25.32 
2 24.64 
2 24.06 
2 23.60 


2 23.60 
2 23.26 
2 23.05 
2 22.99 




XI* 


X* 


IX* 


VIII. 


VII* 


VI* 





TABLE XXIV. 

Sun's Semi-diameter. 
Argument. Sun's Mean Anomaly. 



o 


10 

20 
30 


O* 


I* 


lis 


III* 


IVs 


V* 




16 17.3 
16 17.0 
16 16.2 
16 15.0 


16 15.0 
16 13.3 
16 11.2 
16 8.8 


16 8.8 
16 6.2 
16 3.4 
16 0.6 


16 0.6 
15 57.8 
15 55.1 
15 52.7 


15 52.7 
15 50.5 

15 48.6 
15 47.0 


15 47.0 
15 45.9 
15 45.2 
15 45.0 


o 
30 
20 
10 




XI, 


X, 


IXs 


VIII* 


VII* 


Yh 





ia 



TABLE XXV. 

Equation of the Surfs Centre. 
Argument. Sun's Mean Anomaly. 





08 


Is 


II, 


III, 


TVs 


v. 1 




1 

2 
3 

4 
5 


8 O ' '« 

11 29 59 13.9 
1 17.3 
3 20.6 
5 23.9 
7 27.0 
9 30.0 


57 58.5 

59 43.9 

1 1 28.0 
1 3 10.9 

1 4 52.6 
1 6 33.0 


1 40 10.7 
1 41 8.9 
1 42 5.1 
1 42 59.3 
1 43 51.8 
1 44 42.1 


3 / // 

1 54 34.1 

1 54 30.5 
1 54 24.3 
1 54 17.0 
1 54 7.1 
1 53 55.2 


1 
1 
1 
1 
1 


' " ?' " 

38 4.8 55 52.6 
37 2.4 54 8.7 

35 53.1 52 24.0 
34 52.2 50 33.2 
33 44.6 43 51.6 
32 35.4 47 4.2 


6 
7 

8 

9 

10 


11 32.8 
13 35.4 
15 37.7 
17 39.6 

19 41.2 


1 S 12,3 
1 9 50.1 
1 11 26.5 
1 13 1.7 
1 14 35.3 


1 45 30.4 
1 46 16.3 
1 47 1.2 
1 47 43.5 
1 43 23.9 


1 53 41.0 
1 53 24.9 
1 53 6.7 

1 52 46.5 
1 52 24.2 


1 
1 
1 
1 
1 


31 24.4 
30 11.9 

23 57.7 
27 42.0 
26 24.3 


45 16.0 
43 26.9 
41 37 

39 46 5 
37 55.3 


11 
12 

13 
14 
15 


21 42.4 
23 43.1 
25 43.4 
27 43.2 
29 42.3 


1 16 7.5 
1 17 3?. 2 
1 19 7.5 
1 20 35.2 
1 22 1.5 


1 49 2.2 
1 49 33.4 
1 50 12.6 
1 50 44.7 
1 51 14.9 


1 51 59.8 
1 51 33.4 
1 51 5.0 
1 50 34.5 
1 50 2.2 


1 
1 
1 
1 

1 


25 5.9 
23 45.7 
22 23.8 
21 0.6 
19 36.0 


36 3.3 

34: 
32 I 
30 23.8 
28 2 


16 
17 
18 
19 
20 


31 40.9 
33 38.9 
35 36.2 
37 32.9 
39 23.3 


1 23 26.0 
1 24 48.9 
1 26 10.3 
1 27 30.0 
1 28 48.0 


1 51 42.9 
1 52 S.7 
1 52 32.5 
1 52 54.3 
1 53 13.9 


1 49 27.7 
1 48 51.3 
1 48 13.0 
1 47 32.7 
1 46 50.4 


1 
1 
1 

1 
1 


15 9.9 

16 42.4 

15 13.7 
13 43.5 

12 12.1 


23 34.3 
242 
22 43 .9 
20 47.9 ' 
1851.4 


21 
22 
23 
24 
25 


41 23.9 
43 18.1 

45 11.5 
47 4.0 
43 55.6 


1 30 4.2 
1 31 18.8 
1 32 31.7 
1 33 42.7 
1 34 52.0 


1 53 31.4 
1 53 46.8 
1 54 0.1 

1 54 11.2 
1 54 20.4 


1 46 6.3 
1 45 20.3 
1 44 32.2 
1 43 42.4 
1 42 50.7 


1 
1 

1 
1 
1 


10 39.3 -0 16 54.6 

9 5.4 14 57.5 
7 30.3 13 0.1 
5 54.0 11 2.6 
4 16.5 9 4.3 


26 

27 

2S 

1 29 

! 30 


50 46.3 
52 36.0 

54 24.6 
56 12.1 
57 58.5 


1 35 59.4 
1 37 5.1 
1 33 8.8 
1 39 10.8 
1 40 10.7 


1 54 27.2 
1 54 32.1 
1 54 34.9 
1 54 35.4 
1 54 34.1 


1 41 57.1 
1 41 1.7 
1 40 4.5 
1 39 5.6 
1 33 4.8 


1 

1 



) 


2 37.3 

53.0 

59 17.3 

57 35.4 

55 52.6 


7 6.9 
5 87 
3105 
112.2 

1 



TABLE XXVI. 

Secida?' Variation of Equation of Sim's Centre. 

Argument. Sun's Mean Anomaly. 



1 


O 


I, 


II, 


III* 


IV, 


■ 

v. 


z 






r> 


t. 


// 







— 


— 9 


— 15 


— 17 


— 15 


— 8 


2 


1 


' 9 


15 


17 


14 


8 


4 


1 


10 


16 


17 


14 


7 


6 


2 


10 


16 


17 


14 


7 


S 


2 


11 


16 


17 


13 


6 


10 


3 


11 


16 


17 


13 


6 


12 


4 


12 


17 


17 


12 


5 


14 


4 


12 


17 


16 


12 


5 


16 


5 


13 


17 


16 


12 


4 


18 


5 


13 


17 


16 


11 


3 


20 


6 


13 


17 


16 


11 


3 


22 


7 


14 


17 


16 


10 


2 


24 


7 


14 


17 


15 


10 


2 


26 


8 


15 


17 


15 


9. 


1 


23 


8 


15 


17 


15 


9 


1 


30 


— 9 


— 15 


— 17 


— 15 


— S 


— 



TABLE XXV. 

Equation of the Sun's Centre* 
Argument. Sun's Mean Anomaly. 



19 





Vis 


VII* 


VIIIs 


IXs 


Xs 


XI* 1 




11* 


Us 


lis 


lis 


lis 


11. 


o 


O ' " 


O ' " 


O ' " 


o 


, ,, 


O ' " 


O ' " 





29 59 13.9 


29 2 35.2 


28 20 23.0 


28 


3 53.7 


28 18 17.1 


29 29.3 


1 


29 57 15.6 


29 52.4 


23 19 22.2 


28 


3 52.3 


28 19 17.0 


29 2 15.7 


2 


29 55 17.3 


28 59 10.5 


28 18 23.3 


28 


3 52.8 


28 20 19.0 


29 4 3.2 


3 


29 53 19.1 


2S 57 29.8 


28 17 26.1 


28 


3 55.6 


28 21 22.7 


29 5 51.8 i 


4 


29 51 20.9 


28 55 50.0 


28 16 30.7 


28 


4 0.5 


28 22 28.4 


29 7 41.5 | 


5 


29 49 23.0 


23 54 11.4 


28 15 37.1 


28 


4 7.4 


28 23 35.8 


29 9 32.2 


6 


29 47 25.2 


28 52 33.8 


23 14 45.4 


28 


4 16.6 


28 24 45.1 


29 11 23.S | 


7 


29 45 27.7 


28 50 57.5 


23 13 55.6 


28 


4 27.7 


28 25 56.1 


29 13 16.3 ! 


8 


29 43 30.3 


28 49 22.4 


23 13 7.5 


28 


4 41.0 


28 27 9.0 


29 15 9.7 


9 


29 41 33.2 


2S 47 48.5 


28 12 21.5 


28 


4 56.4 


28 28 23.6 


29 17 3.9 


10 


29 39 36.4 


28 46 15.7 


28 11 37.4 


28 


5 13.9 


28 29 39.8 


29 18 59.0 


11 


29 37 39.9 


28 44 44.3 


28 10 55.1 


28 


5 33.5 


28 30 57.8 


29 20 54.9 


12 


29 35 43.9 


28 43 14.1 


28 10 14.8 


2S 


5 55.3 


28 32 17.5 


29 22 51.6 


13 


29 33 48.2 


23 41 45.4 


28 9 36.5 


28 


6 19.1 


28 33 38.9 


29 24 48.9 


14 


29 31 53.0 


23 40 17.9 


28 9 0.0 


28 


6 44.9 


28 35 1.8 


29 26 46.9 


15 


29 29 58.2 


28 33 51.8 


28 8 25.6 


28 


7 12.9 


' 28 36 26.3 


29 28 45.5 


16 


29 28 4.0 


28 37 27.2 


28 7 53.2 


28 


7 43.1 


28 37 52.6 


29 30 44.6 


17 


29 26 10.1 


28 36 4.0 


28 7 22.8 


28 


8 15.2 


28 39 20.3 


29 32 44-4 


18 


29 24 17.0 


28 34 42.1 


28 6 54.4 


28 


8 49.4 


28 40 49.6 


29 34 44.7 


19 


29 22 24.5 


28 33 21.9 


28 6 28.0 


28 


9 25.6 


28 42 20.3 


29 36 45.4 


20 


29 20 32.5 


28 32 3.0 


2S 6 3.6 


28 10 3.9 


28 43 52.5 


29 38 46.6 


21 


29 18 41.3 


28 30 45.8 


28 5 41.4 


28 10 44.3 


28 45 26.1 


29 40 4S.2 \ 


22 


29 16 50.8 


28 29 30.1 


28 5 21.1 


28 11 26.6 


28 47 1.3 


29 42 50.1 i 


23 


29 15 0.9 


23 2S 15.9 


28 5 2.9 


28 12 11.0 


28 48 37.7 


29 44 52.5 


24 


29 13 11.8 


28 27 3.4 


28 4 46.8 


28 12 57.4 


28 50 15.5 


29 46 55.0 


25 


29 11 23.6 


28 25 52.4 


28 4 32.6 


28 13 45.7 


23 51 54.8 


29 48 57.8 


26 


29 9 36.2 


28 24 43.2 


28 4 20.7 


28 14 36.0 


28 53 35.2 


29 51 0.8 


27 


29 7 49.5 


28 23 35.6 


28 4 10.8 


28 15 28.5 


28 55 16.9 


29 53 3.9 


28 


29 6 3.8 


28 22 29.7 


28 4 3.0 


28 16 22.7 


23 56 59.8 


29 55 7.2 


29 


29 4 19.1 


28 21 25.4 


28 3 57.3 


28 17 18.9 1 28 58 43.9 


29 57 10.5 j 


30 


29 2 35.2 


28 20 23.0 28 3 53.7 28 IS 17.1 | 29 29.3 


29 59 13.9 



TABLE XXVI. 

Secular Variation of Equation of Surfs Centre. 
Argument. Sun's Mean Anomaly. 





Vis 


VII* 


VIIIs 


IXs' 


Xs 


Xls 


o 


„ 


// 


r, 


t* 


// 


> 





+ 


4- 8 


+ 15 


-f 17 


+ 15 


+ 9 


2 


1 


9 


15 


17 


15 


8 


4 


1 


9 


15 


17 


15 


8 


6 


2 


10 


15 


17 


14 


7 


8 


2 


10 


16 


17 


14 


7 


10 


3 


11 


16 


17 


14 


6 


12 


3 


11 


16 


17 


13 


6 


14 


4 


12 


16 


17 


13 


5 


16 


5 


12 


16 


17 


12 


4 


18 


5 


12 


17 


17 


12 


4 


20 


6 


13 


17 


16 


11 


3 


22 


6 


13 


17 


16 


11 


2 


24 


7 


14 


17 


16 


10 


2 


26 


7 


14 


17 


16 


10 


1 


28 


8 


14 


17 


15 


9 


1 


30 


+ 8 


I + 15 


+ 17 


+ 15 


+ 9 


+ o 



20 



TABLE XXVII. 



Nutations. 
Argument. Supplement of the Node, or N. 



Solar Nutation, 



N. 


Long. 


R. Asc. 


Obliq. 


' N. 


Long. 


R. Asc. 


Obliq. 




Long. 


Obliq. 1 




,, 


n 


« 






~/7 


// 











+ 0.0 


+0.0 


+ 9.2 


500 


— 0.0 


— 0.0 


— 9.3 


Jan. 


» 


// 


10 


1.0 


1.0 


9.1 


510 


1.1 


1.0 


9.3 


1 


+ 0.5 


— 0.5 


20 


2.1 


2.1 


9.1 


520 


2.2 


2.0 


9.3 


11 


0.8 


0.4 


30 


3.2 


3.0 


9.0 


530 


3.3 


2.9 


9.2 


21 


1.1 


0.2 


40 


4.2 


4.0 


8.9 


540 


4.4 


3.9 


9.0 


31 


1.2 


— 0.1 


50 


+ 5.2 


+ 4.9 


+ 8.7 


550 


— 5.5 


— 4.8 


— 8.9 


Feb. 






60 


6.2 


6.0 


8.5 


560 


6.5 


5.7 


8.7 


10 


1.2 


+ 0.1 


70 


7.2 


6.9 


s.a 


570 


7.5 


6.6 


8.4 


20 


1.0 


0.3 


80 


8.2 


7.8 


8.1 


580 


8.5 


7.5 


8.1 


March. 
, 2 
12 






90 


9.1 


8.7 


7.8 


590 


9.5 


8.4 


7.8 


0.7 
+ 0.3 


0.4 
0.5 


100 


+ 10.0 


+ 9.4 


+ 7.5 


600 


— 10.4 


— 9.1 


— 7.5 


110 


10.8 


10.3 


7,1 


610 


11.2 


9.9 


7.1 


22 


— 0,1 


0.5 


120 


11.6 


11. 1 


6.7 


620 


12.0 


10.6 


6.7 


April. 

1 

11 

21 






130 


12.4 


11.7 


6.3 


630 


12.8 


11.4 


6.3 


0.5 
0.8 
1.1 


0.5 
0.2 
0.2 


140 


13.1 


12.4 


5.9 


640 


13.5 


12.0 


5.9 


150 


+ 13.8 


+ 13.0 


+ 5.5 


650 


— 14.2 


— 12.6 


— 5.4 


160 


14.4 


1&.6 


5.0 


660 


14.8 


13.2 


4.9 


May. 

1 

11 

21 

31 






170 


15.0 


14.1 


4.5 


670 


15.3 


13.8 


44 


1.2 
1.2 
1.1 
0.8 


+ 0.1 

— 0.1 

0.3 

0.4 


180 


15.5 


14.5 


4.0 


689 


15,8 


14.2 


3.9 


190 


15.9 


14.8 


3.5 


690 


16.2 


14.7 


3.3 


200 


+ 16.3 


+ 15.1 


+ 2.9 


700 


— 16.6 


— 15.0 


— 2.8 


210 


16.6 


15.4 


2.4 


710 


16.9 


15.3 


2.2 


June. 
10 
20 
30 






220 


16.9 


15.6 


1.8 


720 


17.1 


15.4 


1.6 


0.4 
— 0.0 

+ 0.4 


0.5 
0.5 
0.5 


230 


17.1 


15.7 


1.2 


730 


17.2 


15.7 


1.1 


240 


17.2 


15.9 


0.7 


740 


17.3 


15.9 


— 0.5 


250 


+ 17.3 


+ 15.9 


+ 0.1 


750 


— 17.3 


— 15.9 


+ 0.1 


260 


17.3 


15.9 


— 0.5 


760 


17.2 


15.9 


0.7 


July. 
10 
20 
30 


0.7 
1.0 
1.2 


0.4 

0.3 

— 0.1 


270 


17.2 


15.7 


l.l 


770 


17.1 


15.7 


1.2 


280 


17.1 


15.6 


1.6 


780 


I&.9 


15.4 


1.8 


290 


16.9 


15.4 


2.2] 


790 


16.6 


15.3 


2.4 


300 


+ 16.6 


+ 15.1 


— 2.8j 


800 


— 16.3 


— 15.0 


+ 2.9 


Aug. 






310 


16.2 


14.8 


3.3J 


810 


15.9 


14.7 


3.^ 


9 
19 
29- 


1.3 

1.2 
0.9 


+ 0.0 
0.4 
0.4 


320 


15.8 


•14.5 


3.9 


820 


15.5 


14.2 


4.0 


330 


15.3 


14.1 


4.4 


830 


15.0 


13.8 


4.5 


340 


14.8 


13.6 


4.9 


840 


14.4 


13.2 


5.0 


Sept. 






350 


+ 14.2 


4. 13.0 
12.4 


_ 5.4 


850 


_ 13.8 


— 12.6 


+ 5.5 


8 


0.6 


0.5 


360 


13.5 


5.9 


860 


13.1 


12.0 


5.9 


18 
28 


+ 0.2 
— 0.2 


0.5 
0.5 


370 


12.8 


11.7 


6.3 


870 


12.4 


11.4 


6.3 


380 


12.0 


11.1 


6.7 


880 


11.6 


10.6 


6.7 


Oct. 






390 


11.2 


10.3 


7.1 


890 


10.8 


9.9 


7.1 


8 


0.6 


0.5 


400 


+ 10.4 


+ 9.4 


— 7.5 


900 


— 10.0 


— 9.1 


+ 7.5 


18 


1.0 


0.3 


410 


9.5 


8.7 


7.8 


910 


9.1 


8.4 


7.8 


28 


1.2 


0.2 


420 


8.5 


7.8 


8.1 


920 


8.2 


7.5 


8.1 


Nov. 






430 


7.5 


6.9 


8.4 


930 


7.2 


6.6 


8.3 


7 


1.2 


+ 0.0 


440 


6.5 


6.0 


8.7 


940 


6.2 


5.7 


8.5 


17 


1.2 


0.2 


450 


+ 5.5 


+ 4.9 


_ 8.9 


950 


— 5.2 


— 4.8 


+ 8.7 


27 


1.0 


0.4 


460 


4.4 


4.0 


9.0 


960 


4.2 


3.9 


8.9 


Dec. 






470 


3.3 


3.0 


9.2 


970 


3.2 


2.9 


9.0 


7 


0.6 


0.5 


480 


2.2 


2.1 


9.3 


980 


2.1 


2.0 


9.1 


17 


— 0.2 


0.5 


490 


l.r 


1.0 


9.3 


990 


1.0 


1.0 


9.1 


27 


+ 0.3 


0.5 


500 


+ 0.0 


+ 00 


— 9.3 


1000 


— 0.0 


— 0.0 


+ 9.2 


37 


+ 0.6 


— 0.5, 



TABLE XXVIII. 



TABLE XXIX 



2J 



Lunar Equation, \si petit. 
Argument I. 



Lunar Equation, 2d part. 
Arguments I. and VL 
T. 



I Equa I Equ 



7.5 

8.0 
8.4 
8.9 
9.4 
9.8 

10.3 
10.7 
11.1 
11.5 
11.9 

12.3 

12.6 
13.0 
13.3 
13.6 

13.8 
14.1 
14.3 
14.5 
14.6 

14.8 
14.9 
14.9 
15.0 
15.0 

15.0 
14.9 
14.9 
14.S 
14.6 

14.5 
14.2 
14.1 
13.8 
13.6 

13.3 
13.0 
12.0 
12.3 
11.9 

11.5 
11.1 
10.7 
10.3 
9.8 

9.4 
8.9 

84 



,490! 80 
500 ] 7.5 



500 7.5 

510 7.0 

520 6.6 

530 6.1 

540 5.6 

550 5.2 

560 4.7 

570 4.3 

580 3.9 

590 3.5 

600 3.1 

610 2.7 
620 j 2.4 
630 2.0 
640 i 1.7 
650 ! 1.4 



660 
670 
680 
690 
700 

710 
720 



730 | 0.1 
740 I 0.0 
750 0.0 

760 0.0 

770 0.1 

7S0 0.1 

790 0.2 

800 0.4 

810 0.5 
820 0.7 

830 0.9 
840 j 1.2 
850 j 1.4 

860 1 1.7 
870 | 2.0 

8S0 2.4 
890 I 2.7 
900 3.1 

910 3.5 

920 3.9 

930 4.3 

940 4.7 

950 5.2 

960 5.6 

970 6.1 

980 6.6 

990 7.0. 

1000 : 7.5 I 

j 



p 


1 ° 


50 


100 


150 


200 


250 


300 


350 


'400 


450 


500 




" 


„ 


77 


77 


77 


~" 


„ 


77 


"7 


,, 


"77 


1 o 


1.3 


1.2 


1.2 


l.i 


1.0 


1.0 


1.0 


1.1 


1.2 


1.2 


1.3 


50 


1.5 


1.5 


1.5 


1.3 


l.i 


1.0 


0.9 


1.0 


1.1 


1.1 


l.i 


100 


1.7 


1.8 


1.7 


1.4 


1.2 


1.1 


1.0 


0.9 


0.9 


0.9 


0.9 


150 


1.9 


1.9 


1.8 


1.6 1.4 


1.3 


1,0 


0.8 


O.S 


0.8 


0.7 


200 


1.9 


2.0 


2.0 


1.7 1.5 


1.4 


1.0 


O.S 


0.8 


0.8 


0.7 


250 


2*.0 


2.0 


2.0 


1.8 1.6 


1.5 


1.1 


0.9 


0.7 


0.7 


!o.6 


300 


1.9 


1.9 


1 i-o 


1.9 1.7 


1.6 


1:2 


1.0 


0.8 


0.7 


0.7 


350 


1.8 


1.9 


1.9 


1.9 1.7 


1.6 


1.4 


1.0 


1.0 


0.9 


0.8 


400 


1.6 


1.7 


1.8 


1.9 


:.7 


1.6 


1.4 


1.2 


1.1 


1.0 


1.0 


450 


1.5 


1.5 


1.6 


1.7 


1.7 


1.7 


1.6 


1.4 


1.2 


1.2 


1.1 


500 


1.3 


1.4 


1.4 


1.5 


L7 


1.7 


1.7 


1.5 


1.4 


1.4 


1.3 


550 


1.1 


1.2 


1.2 


1.4 


1.6 


1.7 


1.7 


! i7 


1.6 


1.5 


1.5 


600 


1.0 


1.0 


I 1 


1.2 


1.4 


1.6 


1.8 


1.8 


1.8 


1.7 


L6 


650 


0.8 


0.9 


1.0 


1.1 


1.3 


1.5 


1.7 


1.8 


1.0 


1.9 


1.8 


700 


0.7 


0.7 


0.8 


1.1 


1.2 


1.4 


1.7 


1.9 


1 1.9 


1.9 


1.9 


750 


0.6 


0.6 


0.7 


1.0 


1.1 


1.3 


1.6 


1.9 


1.9 


2.0 


2.0 


800 


0.7 


0.7 


0.7 


0.9 


1.1 


1.2 


1.5 


1.8 


2.0 


1.9 


1.9 


850 


0.7 


0.8 


|0.8 


0.9 


0.9 


1.1 


1.4 


1.7 


1.8 


1.8 


1-9 


900 


0.9 


0.9 


0.9 


0.9 


1.0 


1.1 


1.2 


1.5 


1.7 


1.7 


1.7 


950 


1-1 


1.0 


1.1 


1.0 


1.0 


1.0 


1.1 


1.3 


1.4 


1.6 


! 1.5 


o 


1.3 


1.2 


.1.2 


1.1 


1 1.0 


1.0 


1.0 


1.1 


.1.2 


1.2 


1.3 


I. 


VI 


500 


550 


600 


650 


700 


750 


800 


850 


900 


950 1000 




~\ 


"77 




77 


77 


~7~ 


77 


77 


~\ 


" i " 





1.3 


1.4 


1.4 


1.5 


1.6 


1.6 


1.6 


1.5 


1.4 


1.4 1.3 


50 


1.1 


1.1 


1.2 


1.3 


1.5 


1.5 


1.7 


1.6 


1.5 


1.5 1.5 


100 


0.9 


0.9 


0.9 


1.1 


1.3 


1.5 


1.6 


1.7 


1.7 


1.7 1.7 


150 


0.7 


O.S 


0.8 


0.9 


1.2 


1.4 


1.6 


1.9 


1.8 


1.8 


1.9 


200 


0.7 


0.7 


0.6 


0.8 


1.1 


1.2 


1.6 


1.8 


l.S 


1-8 


1.9 


250 


0.6 


0.6 


0.7 


0.7 


1.0 


1.1 


1.5 


1.7 


1.9 


1.9 


2.0 


300 


0.7 j 


0.7 


0.7 


0.7 


0.9 


1.0 


1.4 


1.6 


1.8 ! 


1.9 


1.9 


350 


o.s 


0.7 


0.7 


0.8 


0.9 


1.0 


1.4 


1.6 


1.6 


1.7 


1.8 


400 


1.0 1 


0.9 


O.S 


0.8 


0.9 


1.0 


1.2 


1.4 


1.5 


1.6 


1.6 


450 


1.1 


1.1 


1.0 


0.9 


0.9 


0.9 


1.0 


1.2 


1.4 


1.4 


1.5 


500 


1.3 


1.2 


1.2 


1.1 


0.9 


0.9 


0.9 


1.1 


1.2 


1.2 


1.3 


550 


1.5 


1.4 


1.4 


1.2 


1.0 


0.9 


0.9 


0.9 


U) 


1.1 


1.1 


600 


1.6 


1.6 


1.5 


1.4 


1.-2 


1.0 


0.8 


0.8 


O.S 


0.9 


1.0 


650 


1.8 


1.7 


1.6 


1.6 


1.3 


1.1 


0.9 


0.8 


0.7 1 


0.7 


0.8 


700 


1.9 


1.8 


1.8 


1.6 


1.4 


1.2 


0.9 


0.7 


0.7 1 


0.7 


0.7 


750 


2.0 


1.9 


1.9 


1.7 


1.5 


1.3 


1.0 


0.7 


0.7 


0.6 


0.6 


800 


1.9 


1.8 


1.8 1 


1.8 


1.6 


1.4 


1.1 


0.8 


0.6 


0.7 


0.7 


850 


1.9 


1.8. 


1.8 


1.8 


1.6 


1.5 


1.2 


0.9 


O.S 


0.8 


0.7 


900 


1.7 


1.7 


1.7 


1.7 


1.6 


1.5 


1.3 ! 


1.1 


0.9 


0.9 


0.9 


950 


1.5 


1.5 


1.5 


1.6 


1.7 


1.6 


1.5 


1.3 


1.2 1 


1.1 


1.1 





1.3 


1.4 


1.4 


1.5 


1.6| 


1.6 


1.6 1 


1.5 


1.4 


1.4 


1.3 


Constant 1".3. 

i 



22 



TABLE XXX. 



Perturbations produced by Venus. 

Arguments II and III. 

HI. 



II. 





10 


20 


30 


40 


50 


60 


70 

14.7 


80 


90 


100 


110 


120' 


0;21.6 


20.8 


19.8 


19.0 


17.9 


16.8 


15.9 


14.0 


13.2 


128 


12.5 


12.2 ! 


20 !23.1 


22.7 


21.6 


21.0 


20.1 


19,3 


18.4 


17.4 


16.4 


15.5 


14.5 


13.8 


13.4 l 


40 23.5 


23.2 


22.9 


22.7 


22.0 


21.1 


20.4 


19.5 


18.7 


17.9 


16.9 


16.1 


15.3 | 


1 60 ! 22.2 


22.5 


23J 


22.7 


22.8 


22.5 


21.9 


21.3 


20.5 


19.9 


19.1 


18.2 


17.4 


SO : 20.0 


20.7 


21.4 


21.7 


22.1 


22.3 


22.2 


22.2 


21.7 


21.3 


20.7 


19.9 


19.3 j 


! 200 


17.6 


18.6 


19.2 


19.9 


20.5 


21.0 


21.6 


21.7 


21.6 


21.6 


21.5 


21.1 


20.5 


120 


15.3 


10. 


16.9 


17.7 


1S.4 


19.2 


19.8 


20.2 


20.7 


20.8 


21.1 


21.1 


20.8 


! 140 


13.6 


14.2 


14.8 


15.5 


16.2 


17.0 


17.6 


18.3 


19.0 


19.4 


20.0 


20.0 


20.4 


1 160 


12.7 


13.2 


13.6 


14.1 


14.6 


15.0 


15.7 


16.4 


17.0 


17.3 


18.1 


18.7 


19.2 


! 180 


12.7 


12.9 


13.1 


13.5 


13.9 


14.0 


14.5 


14.8 


15.0 


15.8 


16.4 


16.8 


17.2 


! 200 


13.2 


13.2 


13.2 


13.4 


13.7 


13.8 


14.1 


14.2 


14.5 


14.5 


14.8 


15.2 


16.0 


| 220 


13.5 


13.6 


13.9 


14,1 


14.1 


14.1 


14.2 


14.3 


14,5 


14.6 


14.6 


14.7 


14.8 


1 240 


13.6 


13.8 


14.1 


14.4 


14.6 


14.8 


14.8 


14.9 


15.1 


15.1 


15.1 


14.9 


14.8 


1 260 


12.8 


13.3 


13.8 


14.2 


14.6 


15.0 


15.3 


15.6 


15.5 


15.5 


15.6 


15.6 


15.6 


! 280 


11.5 


12.3 


13.0 


13.4 


14.0 


14,6 


15.1 


15.4 


16.0 


16.2 


16.2 


16.3 


16.2 


! 300 


10.1 


10.9 


11.3 


12.1 


12.9 


13.7 


14,2 


14.9 


15.4 


16.0 


16.4 


16.5 


16.7 


320 


8.2 


8.8 


9.6 


10.6 


11.3 


12.0 


12.9 


13.7 


14.3 


15.0 


15.8 


16.3 


16.8 


340 


6.9 


7.5 


8.1 


8.4 


9.4 


10.1 


11.1 


11.9 


12.7 


13.6 


14.4 


15.2 


16.0 


360 


6.5 


6.5 


6.S 


7.4 


8.0 


8.4 


9.1 


9.9 


10.8 


11.5 


12.6 


13.4 


14.4 


3S0 


6.8 


6.5 


6.3 


6.4 


6.7 


7.0 


7.6 


8.2 


8.9 


9.6 


10.6 


11.4 


12.4 


400 


7.5 


7.1 


6.7 


6.4 


6.2 


6.4 


6.5 


6.9 


7.5 


7.9 


8.7 


9.4 


10.3 


420 


9.1 


8.4 


7.6 


7.1 


6.7 


6.5 


6.3 


6.2 


6.7 


6.8 


7.2 


7.8 


8.4 


440 


10,0 


9.8 


'9.0 


8.6 


7.9 


7.2 


6.7 


6.4 


6.4 


6.4 


6.6 


6.8 


7.1 


460 


12.1 


11.5 


10.5 


9.6 


9.0 


8.5 


8.0 


7.3 


6.8 


6.6 


6.5 


6.4 


6.5 


480 ' 13.6 


12.8 


11.9 


11.0 


10.4 


9.6 


8.S 


8.2 


7.7 


7.2 


6.8 


6.4 


6.5 


500 


15.1 


14.4 


13,4 


12.4 


11.6 


10.8 


10.1 


9.3 


8.6 


8.1 


7.5 


7.1 


6.8 


520 


16.5 


15.fi 


14.8 


13.9 


13.1 


12.3 


11.3 


10.5 


9.7 


9.1 


8.6 


7.9 


7.4 


540 


18.1 


17.5 


16.4 


15.5 


14.5 


13.7 


12.8 


11.8 


11.1 


10.4 


9.7 


8.9 


8.2 


560 


20.4 


19.3 


18.2 


17.6 


16.5 


15.4 


14.4 


13.4 


12.7 


11.6 


10.8 


10.2 


9.2 


580 


22.8 


21.7 


20.7 


19.7 


18.4 


17.6 


16.6 


15.5 


14,3 


13.4 


12.5 


11.6 


10.6 


600 


25.2 j 24.1 


23.1 


22.2 


21.2 


19.9 


18.6 


17.8 


16.6 


15.6 


14.5 


13.4 


12.6 


620 


27.3 20.5 


25.6 


24.7 


23.5 


22.5 


21.6 


20.4 


19.0 


18.1 


16.8 


15.7 


14.7 


640 


29.0 28.5 


27.7 


26.9 


26.2 


25.1 


24.1 


22.9 


21.8 


20.8 


19.6 


18.4 


17.2 


660 


20.8 29.6 


29.2 


28.5 


28.1 


27.4 


26.5 


25.6 


24.5 


23.4 


22.5 


21.2 


19.8 


680 


29.7 29.6 


29.5 


29.5 


29.1 


28.8 


28.2 


27.6 


27.0 


26.0 


25.0 


23.8 


22.8 


700 


28.8 29.2 


29.3 


29.5 


29.5 


29.5 


29.2 


28.8 


28.4 


27.8 


27.2 


26.4 


25.2 


720 


26.9* 27.6 


28.3 


29.0 


29.2 


29.4 


29.4 


29.3 


29.1 


28.9 


28.4 


27.9 


27.3 


740 


24.7 25.7 


26.6 


27.3 


27.9 


28.5 


29.1 


29.0 


29.2 


29.3 


29.1 


28.8 


28.4 


760 


22.2 23.5 


24.3 


25.3 


26.2 


27.0 


27.6 


28.3 


28.6 


28.7 


28.9 


29.1 


29.0 


780 


19.6 21.0 


22.0 


23.2 


24.2 


25.1 


25.9 


26.7 


27.3 


27.8 


28.4 


28.5 


28.7 


800 


17.2 18.5 


19.3 


20.9 


21.8 


22.9 


23.9 


25.0 


25.8 


26.4 


26.9 


27.6 


28.1 


820 


15.2 15.9 


17.0 


18.4 


18.9 


20.7 


21.7 


22.8 


23.8 


24.8 


25.6 


26.2 


26.6 


840 


13.2 14.0 


15.0 


10.0 


17.0 


18.2 


18.8 


20.3 


21.7 


22.7 


23.6 


24.5 


25.3 


860 


11.5 12.2 


13.0 


13.9 


14.9 


15.9 


'17.1 


18.0 


18.9 


20.3 


21.4 


22.6 


23.5 


880 


11.0 11.2 


11.5 


12.2 


13.0 


13.7 


14.8 


15.7 


16.8 


18.1 


19.1 


20.2 


21.1 


900 


11.2 10.2 


10.9 


11.5 


12.5 


12.1 


12.8 


13.7 


14.5 


15.5 


16.6 


17.9 


18.5 


920 


12.1 11.6 


11.5 


11.1 


11.2 


11.3 


11.7 


12".l 


12.7 


13.4 


14.4 


15.2 


16.4 


940 


14.0 13.3 


12.6 


12.3 


11.6 


11.5 


11.3 


11.4 


11.6 


12.0 


12.8 


13.3 


14.2 


960 


16.7 15.6 


14.6 


1-3.7 


13.1 


12.5 


11.9 


11.7 


11.6 


11.4 


11.7 


12.1 


12.6 


980 


19.5 18.3 


17.3 


16.4 


15.2 


14.2 


13.4 


12.7 


12.2 


12.0 


11.9 


11.8 


11.8 


1000 


21,6^0.8 


19.8 


19.0 


17.9 


16.8 


15.9 


14.7 


14.0 


13.2 


12.8 


12.5 


12.2 







10 


20 


30 


40 


50 


60 


70 


80 


90 


00 


110 


120 



TABLE XXX. 



23 



Perturbations produced by Venus, 

Arguments II and III. 

III. 



II. 


i 120 


130 


1 140 


150 


160 


170 
13.3 


180 
13.9 


190 


j 200 


210 


220 


230 


240 





12.2 


12.2 


12.3 


12.4 


12.8 


14.7 15.6 


16.5 


17.7 


18.8 


20.1 


20 


13.4 


12.9 


12.6 


12.3 


12.2 


12.4 


12.9 


13.3 


14.0 


14.6 


15.5 


16.4 


17.3 


40 


15.3 


14.4 


14.0 


13.5 


'13.0 


12.9 


12.6 


12.6 


13.1 


13.5 


14.0 


14.4 


15.4 


00 


17.4 


16.7 


16.0 


15.2 


14.5 


14.0 


13.6 


13.3 


13.2 


13.2 


13.4 


13.5 


14.1 


80 


j 19.3 


; IS. 7 


17.7 


17.1 


16.4 


15.9 


15.4 


14.6 


1 14.3 


13.9 


13.8 


13.7 


13.6 


100 


20.5 


20.2 


19.5 


18.9 


! 18.2 


17.5 


17.1 


16.3 


15.9 


15.4 


14.8 


14.6 


14.3 


120 


20.8 


1 20.7 


20.4 


20.0 


19.7 


19.2 


18.5 


1S.0 


17.3 


16.9 


16.5 


16.2 


15.6 


14-0 


20.4 


; 20.4 


20.2 


20.0 


20.1 


19.7 


19.5 


19.3 


18.8 


18.2 


17.7 


17.4 


17.0 


160 


19.2 


19.1 


19.4 


19.7 


19.5 


19.6 


19.3 


19.6 


19.2 


19.0 


18.7 


18.4 


18.1 


180 


17.2 


\ 17.7 


18.5 


18.5 


18.5 


18.8 


18.4 


1S.8 


19.0 


19.0 


1S.9 


18.6 


18.5 


200 


16.0 


16.2 


16.6 


16.8 


17.5 


17.6 


17.7 


17.9 


18.1 


18.2 


18.3 


18.3 


18.3 


220 


14.8 


15.Q 


15.3 


15.7 


16.1 


16.2 


16.6 


16.8 


17.1 


17.5 


17.1 


17.4 


17.5 


240 


14.8 


14.7 


14.S 


15.0 


15.1 


15.4 


15.7 


15.8 


16.0 


16.1 


16.1 


16.3 


16.4 


260 


15.6 


15.7 


15.3 


14.8 


15.0 


15.0 


15.1 


15.0 


15.1 


15.2 


15.2 


15.1 


15.3 


280 


16.2 


16.2 


16.2 15.9 


15.8 


15.8 


15.5 


15.4 


15.1 


14.9 


14.8 


14.7 


15.0 


300 


16.7 


17.0 


17.1 


16.9 


16.9 


16.6 


16.5 


16.3 


15.9 


15.7 


15.2 


14.9 


14.8 


320 


16.8 


17.3 


17.5 


17.6 


17.7 


17.6 


17.5 


17.2 


17.0 


16.8 


16.5 


16.1 


15.6 


340 


16.0 


16.4 


17.2 


17.8 


17.9 


18.1 


18.3 


18.2 


18.2 


17.9 


17.5 


17.3 


16.8 


360 


14.4 


15.2 


16.0 


16.7 


17.4 


18.1 


18.4 


18.6 


18.8 


18.8 


18.8 


18.7 


18.4 


380 


12.4 


13.4 


14.3 


15.3 


16.1 


16.9 


17.5 


18.1 


18.6 


19.1 


19.3 


19.5 


19.5 


400 


10.3 


11.2 


12.3 


13.2 


14.2 


15.1 


16.0 


16.8 


17.8 


18.4 


18.8 


19.3 


19.8 


420 


8.4 


9.2 


10.0 


11.0 


12.2 


13.0 


14.1 


15.0 


15.9 


16.9 


17.7 


18.5 


19.0 


440 


7.1 


7.6 


8.4 


9.0 


9.9 


10.9 


11.8 


12.9 


13.8 


14.9 


16.0 


16.7 


17.8 


460 


6.5 


6.8 


7.2 


7.4 


8.1 


9.0 


9.7 


10.6 


11.7 


12.6 


13.8 


14.6 


15.9 


480 


6.5 


6.5 


6.4 


6.6 


7.0 


7.5 


8.2 


8.8 


9.6 


10.4 


11.5 


12.5 


13.5 


500 


6.8 


6.7 


6.5 


6.3 


6.5 


6.6 


7.0 


7.4 


8.2 


8.6 


9.4 


10.4 


11.3 


520 


7.4 


7.0 


6.8 


6.5 


6.3 


6.1 


6.3 


6.6 


7.0 


7.5 


8.0 


8.8 


9.3 


540 


8.2 


7.6 


7.2 


6.8 


6.5 


6.3 


6.2 


6.0 


6.2 


6.5 


6.9 


7.4 


7.9 


560 


9.2 


8.6 


7.9 


7.5 


6.8 


6.6 


6.3 


6.1 


6.0 


6.1 


6.2 


6.5 


6.9 


5S0 


10.6 


9.8 


9.1 


8.4 


7.7 


7.3 


6.6 


6.3 


6.1 


5.9 


5.7 


5.9 


6.0 


600 


12.6 


11.4 


10.5 


9.5 


8.7 


8.1 


7.4 


7.0 


6.4 


6.1 


5.8 


5.5 


5.6 


620 


14.7 


13.5 


12.4 


11.4 


10.4 


9.5 


8.7 


7.9 


7.3 


6.7 


6.2 


5.6 


5.2 


040 


17.2 


16.2 


14.9 


13.7 


12.5 


11.4 


10.4 


9.5 


8.7 


7.8 


7.0 


6.5 


5.9 


660 


19.8 


19.0 


17.6 


16.5 


15.1 


13.9 


12.8 


11.5 


10.5 


9.6 


8.6 


7.7 


6.9 


680 


22.8 


21.7 


20.4 


19.3 


18.1 


16.8 


15.7 


14.2 


13.0 


11.9 


10.7 


9.6 


8.6 


TOO 


25.2 


24.3 


23.3 


22.1 


20.7 


19.7 


18.5 


17.3 


16.0 


14.3 


13.4 


12.1 


11.0 


720 


27.3 


26.4 


25.7 


24.5 


23.7 


22.5 


21.1 


20.2 


18.8 


17.7 


16.4 


15.3 


13.9 


740 


28.4 


27.7 


27.4 


26.6 


25.9 


24.9 


24.0 


22.8 


21.5 


20.6 


19.2 


18.1 


16.8 


760 


29.0 


2S.7 


28.3 


27.8 


27.3 


26.8 


25.9 


25.2 


24.3 


23.0 


21.7 


20.7 


19.7 


780 


28.7 


28.7 


28.8 


28.7 


28.3 


28.0 


27.2 


26.1 


20.1 


25.2 


24.3 


23.3 


22.2 


800 


28.1 


28.3 


28.4 


28.5 


28.5 


28.4 


28.2 


27.3 


27.3 


26.7 


25.9 


25.1 


24.4 


820 


26.6 


27.3 


27.8 


28.1 


28.3 


28.1 


28.1 


28.0 


27.9 


27.7 


27.2 


26.5 


25.9 


840 


25.3 


26.2 


26.7 


27.2 


27.5 


27.9 


28.1 


28.1 


27.9 


27.9 


27.6 


27.3 


27.2 


860 


23.5 


24.5 


25.1 


25.9 


26.6 


27.1 


27.4 


27.7 


27.9 


28.0 


27.9 


27.7 


27.5 


880 


21.1 


22.4 


23.3 


24.2 


25.1 


25.8 


26.5 


27.0 


27.3 


27.5 


27.8 


28.0 


27.7 


900 


18.5 


20.1 


21.3 


22.1 


23.1 


24.7 


25.0 


25.7 


26.3 


26.9 


27.3 


27.5 


27.6 


920 


16.4 


17.7 


18.4 


20.0 


21.0 


22.2 


23.0 


23.9 


24.9 


25.7 


26.2 


26.9 


27.3 


940 


14.2 


14.9 


16.1 


17.5 


18.2 


19.6 


20.8 


21.9 


23.0 


23.9 


24.7 


25.7 


26.1 


960 


12.6 


13.3 


14.1 


14.4 


15.9 


17.2 


17.9 


19.5 


20.5 


21.7 


22.7 


23.9 


24.7 


980 


11.8 


12.1 


12.7 


13.3 


14.1 


14.8 


15.6 


16.8 


17.6 


19.3 


20.2 


21.4 


22.6 


1000 


12.2 


12.2 


12.3 


12.4 


12.8 


13.3 


13.9 
180 


14.7 


15.6 
200 


16.5 


17.6 


18.8 

230 
i 


20.1 




•20 


130 


140 


150 


160 


170 


190 


210 


220 


240 



24 



TABLE XXX. 



Perturbations produced by Venus. 

Arguments II. and III. 

III. 



JJL 


| 240 


250 
21.1 


• 260 


| 270 


2S0 


290 


300 


310 


,320 


330 


340 


350 
27.6 


360 


20.1 


23.4 24.3 


25.2 


25.S 


26.6 


27.2 


27.6 


27.7 


27.6 


20 17.3 


IS. 5 


19.7 


20.9 21.9 


23.0 


24.2 


124.9 


25.8 


26.6 


27.0 


27.4 


27.7 


i 40 15.4 


16.5 


17.3 


j 18.3 19.4 


20.5 


21.6 


22.7 


23.7 


24.9 


25.5 


26.3 


26.9 


60 14.1 


14.6 


15.2 


16.3 17.2 


1S.1 


18.9 


20.3 


21.2 


22.3 


23.4 


24.5 


25.3 


j SO 13.6 


14.0 


14.5 


14.9. 15.5 


16.3 


17.3 


18.2 


19.0 


20.0 


21.1 


22.0 


23.1 


| 100 14.3 


14.3 


14.3 


14.4 14.6 


1 15.0 


15.5 


16.2 


16.9 


17.7 


18.9 


19.8 


20.8 


! 120 15.6 


15.2 


14.8 


14.8 15.0 


14.9 


15.0 


15.2 


15.9 


16.3 


17.0 


17.7 


13.5 


140 1T.0 


16.6 


16.4 


15.S 15.5 


15.4 


15.6 


15.6 


15.5 


15.6 


16.1 


16.7 


17.1 


i 160 


18.1 


17.7 


17.5 


17.3 16.9 


16.6 


16.3 


15.9 


16.1 


16.3 


16.3 


16.2 


16.5 


180 


18.5 


18.5 


1S.3 


18.1 17.9 


17.6 


17.5 


17.3 


17.0 


16.9 


16.7 


16.8 


16.9 


! 200 


j 18.3 


18.4 


18.2 


1S.2 1S.2 


IS. 2 


18.1 


18.1 


17.8 


17.7 


17.6 


17.5 


17.7 


■:■; 


' 17.5 


17.6 


17.8 


17.S 18.0 


18.0 


18.2 


18.1 


18.1 


18.3 


18.4 


18.3 


IS. 3 


j 240 


16.4 


16.5 


16.7 


16.9 17.1 


17.3 


17.3 


17.7 


17.5 


18.0 


18.3 


18.4 


IS. 6 


i 260 


15.3 


15.5 


15.5 


15.6 


15.8 


16.1 


16.4 


16.6 


16.8 


16.9 


17.4 


17.7 


18.2 


2S0 15.0 








14.9 




15.0 


15.3 


15.5 


15.9 


16.1 


16.4 


16.8 


300 14.8 


14.6 


14.6 


14.2 


14.0 


14.0 


13.9 


13.9 


14.2 


14.5 


14.S 


15.0 


15.5 


320 15.6 


15.3 14.7 




14.4 


13.1 


13.6 


13.4 


13.3 


13.1 


13.4 


13.6 


13.S 


340 16. S 


16.6 


16.0 


15.5 


15.2 


14.5 


14.3 


13.7 


13.1 


13.0 


12.7 


12.6 


12.6 


360 


J 18.4 


17.9 


17.5 


17.0 


16.5 


15.9 


15.4 


14.9 


14.3 


13.7 


13.0 


12.6 


12.3 


380 


19.5 


19.2 18.9 


18.5 


17.9 


17.7 


16.9 


16.4 


15.8 


15.0 


14.5 


13.6 


13.1 


400 


19.S 


19.8 20.1 


19.7 


19.4 


19.1 


18.6 


18.1 


17.5 


17.0 


16.1 


15.2 


14.8 


420 


19.0 


19.5 20.0 


20.3 


20.3 


20.3 : 20.1 


19.4 


19.0 


18.9 


18.1 


17.3 


16.5 


440 17.S 


IS. 7 


19.2 


19.7 


20.1 


20.4J 20.7 


20.7 


20.5 


20.2 


19.8 


19.5 


18.6 


460 15.9 


16.S 


17.6 


18.6 


19.2 


19.9 120.3 


20.6 


21.0 


20.9 


20.9 


20.8 


20.3 


4S0 13.5 


14.6 


15.5 


16.6 


17.7 


18.5 i 19.3 


19.9 


20.5 


20.8 


21.1 


21.2 


21.2 


500 


11.3 


12.4 


13.4 


14.4 


15.5 


15.5 


17.7 


1S.6 


19.1 


19.9 


20.7 


21.0 


21.4 


520 


9.3 


10.2 


11.2 


12.2 


13.3 


14.2 


15.4 


16.4 


17.6 


18.4 


19.2 


19.S 


20.6 


540 


7.9 


S.6 


9.4 


10.1 


11.1 


.12.1 


13.1 


14.2 


15.3 


16.3 


17.4 


1S.3 


19.2 


560 


6.9 


7.2 


7.8 


8.4 


9.2 


10.1 


11.0 


11.9 


13.1 


14.1 


15.2. 


16.2 


17.2 


580 


6.0 


6.3 


6.6 


7.0 


7.6, 8.4 


9.1 


9.9 


10.9 


11.9 


12.9 


14.1 


15.0 


600 


5.6 


5.6 


5.8 


6.1 


6.5 6.S 


7.4 8.1 


8.8 


9.9 


10.7 


11.8 


12.8 


620 


5.2 


5.4 


5.3 


5.3 


5.5 5.9 


6.3 6.6 


7.2 


8.0 1 


8.7 


9.5 


10.6 


640 


5.9 


5.6 


5.2 


4.9 


5.0 5.0 


5.2 5.5 


5.8 


6.4 


7.0 


7.6 


8.5 


i 660 


6.9 


6.3 


5.7 


5.4 


5.0 4.8 


4.5 4.7 


4.9 


5.1 


5.5 


6.0 


6.8 


6S0 


8.6 


7.6 


6.9 


6.2 


5.6 5.1 


4.S 


4.6 


4.2 


4.2 


4.5 


4.6 


5.1 


700 


11.0 


10.0 


S.7 


7.8 


6.8 6.3 


5.6 


5.0 


4.6 


4.2 


4.2 


4.0 


4.2 


720 


13.9 


12.5 


11.2 


10.3 


9.1 7.9 


7.1 


6.2 


5.6 


4.8 


4.5 


4.2 


3.S 


740 


16.S 


15.5 


14.4 


13.0 


11.7 10.5 


9.4 


8.4 


7.2 


6.5 


5.6 


5.0 


4 3 


760 


19.7 


18 5 


17.2 


15.9 


14.7 13.5 


12.2 


10.8 


9.8 


S.9 


7.6 


6.7 


5.9 


780 


22.2 


21.2 


20.1 


19.0 


17.61 


16.3 j 


15.1 


14.0 


12.6 


11.6 


10.2 


9.2 


8.1 


800 


24.4 


23.4 


22.2 


21.3 


20.3 


19.21 


18.0 


16.7 


15.41 


14.3 


13.2 


11.9 


10.8 


820 


25.9 


25.1 


24.4 


23.3 


22.3 


21.6 


20.4 


19.4 


18.2 


17.2 


15.9 


14.6 


13.6 


840 


27.2 


26.6 


25.8 


25.0 


24.3 


23.5 


22.4 


21.6 


20.5 


19.4 


1S.4 


17.3 


16.4 


860 


27.5 


27.1 26.8 


26.4 


25.5 


24.8 


24.3 


23.3 


22.2! 


21.5 


20.5 


19.6 


IS. 4 


880 




27.5 27.2 


27.0 


26.5 


26.0 


25.5 


24.7 


24.1 


23.2 


22.0 


21.41 


20.4 


900 


27.6 


27.S 


27.9 


27.6 


27.1 


26.7 


26.5 


25.7 


25.3 


24.6 


23.9 


23.0 


22.0 


920 


27.3 


27.5 27.5 


27.6 


27.7 


27.5 


27.2 


26.7 


26.3 


25.7 


25.1 


24.3 


23.6 


940 


26.1 


26.7 27.2 


27.4 


27.7 


27.7 


27.6 


27.5 27.1 


26.6 


26.2 


25.6 


25.5 


| 960 




25.4 


26.2 


26.6 


27.2 


27.5 


27.7 


27.7 27.6 


27.4 


27.1 


27.0 


26.2 


980 


22 6 


23.7 


24.6 


25.3 


25.9 


26.8 


27.2 27.5 


27.7 


27.8 


27.6 


27.5 


27.1 


1000 


20.1 


21.1 


22.2 


23.4 


24.3 


25.2 


25.8 26.6 


27.2 


27.6 
330 


27.7 


27.6 


27.6 
360 




240 


250 • 


260 1 


270 


280 


290 ! 


300 | 310 j 


320 


340 


330 



TABLE XXX. 



25 



Perturbations produced by Venus. 

Arguments II. and III. 

III. 



In. 




360 
27.6 


370 


380 


390 


400 


410 


420 


430 


440 


450 


460 


470 


480 


27.7 


27.3 


26.7 


26.2 


25.5 


24.7 


23.8 


23.1 


22.3 


21.3 


20.2 


19.3 


20 


27.7 


27.8 


27.8 


27.6 


27.4 


26.8 


26.2 


25.6 


24.8 


24.0 


23.1 


22.0 


20.9 


40 


26.9 


27.3 


27.6 


27.9 


27.9 


27.7 


27.5 


27.1 


26.3 


25.6 


24.9 


24.0 


23.2 


60 


25.3 


26.0 


26.8 


27.1 


27.5 


27.9 


27.8 


27.7 


27.3 


27.1 


26.7 


25.9 


25.0 


80 


23.1 


24.0 


25.1 


25.9 


26.5 


27.3 


27.5 


27.9 


28.2 


28.0 


27.6 


27.5 


27.2 


100 


20.8 


21.8 


22.6 


23.6 


24.6 


25.5 


26.2 


26.7 


27.2 


27.5 


27.6 


27.8 


27.4 


120 


18.5 


19.6 


20.6 


21.5 


22.4 


23.2 


24.1 


25.1 


25.8 


26.4 


26.9 


27.3 


27.5 


140 


17.1 


17.9 


18.6 


19.3 


20.3 


21.3 


22.0 


22.9 


23.7 


24.7 


25.5 


26.0 


26.7 


160 


16.5 


17.1 


17.4 


18.1 


18.8 


19.3 


20.1 


21.0 


21.9 


22.6 


23.5 


24.2 


25.1 


180 


16.9 


17.0 


17.1 


17.4 


18.0 


18.4 


18.9 


19.4 


20.1 


20.7 


21.2 


22.2 


23.0 


200 


17.7 


17.5 


17.7 


17.7 


17.6 


1S.1 


18.3 


18.7 


19.2 


19.7 


20.1 


20.8 


21.5 


220 


18.3 


18.2 


1S.3 


18.3 


18.3 


18.3 


18.6 


18.7 


18.9 


19.3 


19.5 


20.0 


20.4 


240 


18.6 


18.8 


18.9 


18.9 


18.9 


19.0 


19.2 


19.1 


19.2 


19.5 


19.6 


19.7 


19.9 


260 


18.2 


IS. 5 


1S.7 


18.8 


19.0 


19.3 


19.5 


19.6 


19.9 


19.9 


20.0 


20.1 


20.2 


280 


16.8 


17.4 


17.9 


18.3 


18.7 


19.1 


19.3 


19.8 


20.0 


20.2 


20.4 


20.6 


20.8 


300 


15.5 


15.8 


16.2 


16.6 


17.6 


18.1 


18.5 


19.2 


19.4 


19.9 


20.6 


20.8 


20.9 


320 


13.8 


14.2 


14.6 


15.1 


15.6 


16.2 


16.8 


17.7 


18.3 


18.9 


19.5 


20.1 


20.8 


340 


12.6 


12.9 


13.0 


13.3 


13.7 


14.4 


14.9 


15.5 


16.2 


17.1 


18.0 


18.6 


19.4 


360 


12.3 


12.1 


11.9 


12.0 


12.3 


12.5 


13.0 


13.4 


14.2 


14.9 


15.7 


16.5 


17.3 


380 


13.1 


12.5 


11.9 


11.6 


11.5 


11.4 


11.6 


11.7 


12.3 


12.7 


13.3 


14.0 


15.0 


400 


14.S 


13.9 


13.1 


12.5 


11.7 


11.2 


11.1 


10.9 


11.0 


11.1 


11.4 


12.0 


12.6 


420 


16.5 


15.7 


15.1 


14.3 


13.4 


12.5 


11.7 


11.1 


10.8 


10.8 


10.5 


10.6 


10.7 


440 


18.6 


17.9 


17.1 


16.1 


15.6 


14,4 


13.5 


12.8 


11.9 


11.1 


10.6 


10.3 


10.3 1 


460 


20.3 


19.8 


19.3 


18.5 


17.6 


16.8 


15.9 


14.7 


13.7 


12.9 


1,2.0 


11.1 


10.9 


480 


21.2 


21.1 


20.8 


20.3 


19.7 


19.1 


18.3 


17.4 


16.4 


15.0 


14.1 


13.2 


12.2 


500 


21.4 


21.4 


21.4 


21.3 


21.1 


20.8 


20.0 


19.5 


18.8 


17.8 


17.0 


15.7 


14.4 


520 


20.6 


21.2 


21.7 


21.7 


21.5 


21.5 


21.4 


21.1 


20.5 


19.8 


19.1 


18.2 


17.6 


540 


19.2 


20.0 


20.7 


21.1 


21.8 


22.0 


21.8 


21.7 


21.5 


21.2 


20.9 


20.3 


19.6 


560 


17.2 


18.4 


19.0 


20.0 


20.8 


21.1 


22.7 


21.9 


22.2 


22.1 


21.9 


21.7 


21.1 


580 


15.0 


16.0 


17.3 


18.2 


19.1 


19.9 


20.8 


21.1 


21.7 


22.0 


22.2 


22.3 


22.1 


600 


12.8 


13.9 


15.1 


15.9 


17.2 


18.0 


19.0 


19.9 


20.6 


21.3 


21.8 


22.0 


22.4 


620 


10.6 


11.5 


12.7 


13.7 


14.9 


16.0 


17.1 


18.3 


19.1 


19.9 


20.8 


21.3 


22.0 


640 


8.5 


9.5 


10.4 


11.3 


12.3 


13.7 


14.9 


16.0 


17.1 


18.1 


19.0 


19.9 


20.7 


660 


6.8 


7.4 


8.2 


9.1 


10.1 


11.1 


12.2 


13.6 


14.6 


15.8 


17.1 


18.1 


19.0 


680 


5.1 


5.7 


6.4 


7.1 


7.9 


8.7 


9.7 


11.0 


12.1 


13.1 


14.1 


15.7 


16.8 


700 


4.2 


4.4 


4.7 


5.1 


5.8 


6.7 


7.4 


8.4 


9.4 


10.6 


11.5 


13.0 


14.1 


720 


3.8 


3.8 


3.8 


4,0 


4.4 


4.8 


5.4 


5.9 


6.9 


8.0 


9.1 


10.1 


11.5 


740 


4.3 


3.9 


3.8 


3.7 


3.6 


3.8 


3.9 


4.4 


4.9 


5.7 


6.4 


7.4 


8.9 


760 


5.9 


5.1 


4.4 


4.0 


3.6 


3.4 


3.4 


3.5 


3.9 


4.3 


4.7 


5.2 


5.9 


780 


8.1 


7.1 


6.1 


5.3 


4.6 


4.1 


3.7 


3.3 


3.3 


3.1 


3.4 


3.6 


4.1 


800 


10.8 


9.7 


8.5 


7.5 


6.5 


5.6 


4.9 


4.2 


3.8 


3.4 


3.2 


3.1 


3.1 


820 


13.6 


12.5 


11.2 


10.1 


9.0 


8.0 


6.9 


6.1 


5.3 


4.7 


3.9 


3.7 


3.1 


840 


16.4 


15.1 


13.7 


12!9 


11.7 


10.6 


9.5 


8.6 


7.5 


6.6 


5.7 


4.9 


4.4 


860 


18.4 


17.5 


16.6 


15.4 


14.3 


13.1 


12.1 


11.1 


10.0 


9.1 


7.9 


7.0 


6.3 


880 


20.4 


19.6 


18.7 


17.5 


16.6 


15.6 


14.5 


13.6 


12.5 


11.5 


10.4 


9.5 


8.6 


900 


22.0 


21.1 


20.2 


19.4 


18.7 


17.7 


16.5 


15.7 


14.7 


13.8 


12.5 


11.9 


10.9 


920 


23.6 


22.7 


21.7 


21.1 


20.1 


19.4 


18.4 


17.5 


16.7 


15.6 


14.8 


13.9 


13.1 


.940 


25.5 


24.1 


23.4 


22.4 


21.4 


20.6 


19.9 


19.0 


18.2 


17.3 


16.6 


15.7 


14.8 


960 


26.2 


25.6 


24.7 


24.1 


23.3 


22.3 


21.3 


20.6 


19.0 


18.9 


17.9 


17.1 


16.3 


980 


27.1 


26.7 


26.3 


25.5 


24.9 


23.8 


23.4 


22.2 


21.0 


20.4 


19.4 


18.6 


17.7 


1000 


27.6 


27.7 


27.3 


26.7 


26.2 


25.5 


24.7 


23.8 
430 1 


23.1 
440 


22.3 
450 


21.3 
460 


20.2 


19.3 




360 


370 


380 


390 


400 


410 


420 


470 


480 



26 



TABLE XXX. 



Perturbations produced by Venus. 

Arguments II and III. 

III. 



II. 




480 
19.3 


490 
18.3 


500 

17.4 


510 

16.6 


520 
15.7 


530 


540 


550 


560 


570 


580 


590 


! 600 


15.0 


14.2 


13.6 


13.1 


12.3 


11.7 


11.3 


! " 
10.S i 


20 


20.9 


20.2 


19.1 


18.2 


17.1 


16.2 


15.5 


14.7 


14.1 


13.3 


12.7 


12.2 


: 11.5 


40 


23.2 


22.0 


20.8 


20.1 


18.9 


17.9 


17.1 


15.9 


15.1 


14.4 


13.7 


13.0 


1. 


60 


25.0 


i 24.0 


23.2 


22.0 


20.7 


.19.9 18.9 


17.7 


16.8 


15.8 


14.9 


14.0 


13.3 


80 


27.2 


126.4 


25.6 


24.1 


23.2 


22.1 | 20.8 


20.0 


18.7 


17.9 


16.6 


15.6 


! 14,8 


100 


27.4 


27.2 


26.8 


26.3 


25.4 


24.5 23.5 


22.2 


20.9 


20.0 


18.6 


17.6 


16.6 


120 


27.5 


27.5 


27.6 


27.1 


26.8 


26.3 | 25.4 


24.6 


23.7 


22.4 


21.0 


20.1 


18.8 


140 


26.7 


27.0 


27.2 i 27.4 


27.3 


27.4 j 26.9 


■ 26.2 


25.4 


24.6 


23.9 


22.6! 21.1 


160 


25.1 


25.6 


26.1 ! 26.7 


26.9 


27.3 127.1 


27.0 


26.9 


26.4 


25.5 


24.7 


23.9 


180 


23.0 


23.S 


24.5 25.0 


25.7 


26.3 


26.7 


26.8 


27.0 


26.8 


26.6 


26.2 


25.6 


200 


21.5 


22.2 


22.8 | 23.5 


24.1 


24.7 


25.5 


25.8 


26.3 


26.6 


26.6 


26.6 


26.4 


220 


20,4 


21.0 


21.5 


22.0 


22.6 


23.2 


23.8 


24,5 


25.0 


25.4 


25.8 


26.0 


26.2 


240 


19.9 


20.4 


20.8 


21.2 


21.6 


21. S 


22.2 


22.6 


23 1 


23.3 


23.9 


24.2 


24.6 


260 


20.2 


20.3 


20.6 


21.2 


21.4 


21.7 


21.9 


22.2 


22 3 


22.7 


23.1 


23.3 


23.6 


280 


20.8 


20.8 


21.0 


21.1 


21.3 


21.4 


21.5 


21.8 


22.0 


22.2 


22.7 


23.0 


23.3 


300 


20.9 


21.0 


21.5 


21.7 


21.7 


22.0 


22.0 


22.1 


22 1 


22.2 


22 4 


22.6 


22.8 


320 


20.8 


21.2 


21.5 


21.6 


22.0 


22.3 


22.5 


22.5 


22 6 


22.7 


22.8 


22.8 


22.9 


340 


19.4 


20.2 


20.8 


21.5 


21.9 


22.1 


22.6 


23.0 


23.2 


23.4 


23.3 


23.4 


23.5 


360 


17.3 


18.4 


19.5 


20.0 


20.6 


21.5 


22.2 


22.7 


23.0 


23.7 


23.7 


24.0 


24,2 


380 


15.0 


15.9 


16.9 


17.8 


18.6 


19.6 


20.6 


21.5 


22.3 


22.9 


23 5 


23.9 


24.5 


400 


12.6 


13.2 


14.2 


15.4 


16.2 


17.3 


18.3 


19.2 


20.3 


21.4 


22 4 


23.0 


23.7 


420 


10.7 


11.2 


12.0 


12.5 


13.5 


14.5 


15.6 


16.7 


17.7 


18.7 


20 1 


21.0 


22.0 


440 


10.3 


10.2 


10.3 


10.5 


11.3 


12.0 


12.9 


13.6 


14.7 


16.0 


17.0 


18.3 


19.5 


460 


10.9 


10.1 


9.9 


9.9 


9.9 


10.1 


10.7 


11.3 


12.2 


L3.0 


140 


15.1 


16.5 


480 


12.2 


11.4 


10.7 


10.1 


9.7 


9.5 


9.7 


9.9 


10.2 


10.7 


11.7 


L2.5 


13.4 


500 


14.4 


13.6 


12.5 


11.6 


10.9 


10.2 


9.8 


9.4 


9.3 


9.6 


9.8 


10.2 


11.1 


520 


17.6 


16.2 


15.1 


13.9 


12.9 


11.9 


10.9 


10.3 


9.8 


9.5 


9.2 


9.2 


9.6 


540 


19.6 


18.6 


18.0 


16.7 


15.4 


14.5 


12.2 


12.3 


L1.3 


10.5 


10.1 


9.5 


9.3 


500 


21.1 


20.4 


19.8 


19.0 


18.2 


17.2 


16.0 


14.8 


13.7 


12.7 


11.7 


10.9 


10.2 


580 


22.1 


21.8 


21.5 


20.9 


20.3 


19.3 


18.6 


17.3 


16.5 


15.4 


14.0 


12 9 


12.2 


600 


22.4 


22.4 


22.2 


22.2 


21.5 


21.2 


20 6 


19.5 


19.1 


17.7" 


16.8 


]5.8 


14.4 


620 


22.0 


22.3 


22.4 


22.4 


22.3 


22.3 


21 9 


21.5 


20.9 


20.0 


19.3 


18.0 


16.9 


640 


20.7 


21.7 


22.0 


22.3 


22.6 


22.5 


22 6 


22.4 


22.0 


21.6 


21.1 


20 3 


19.6 


660 


19.0 


20.0 


20.8 


21.3 


22.1 


22.3 


22 6 


22.8 


22.7 


22.6 


22.2 


21.8 


21.3 


680 


16.8 


18.0 


L9.0 


19.9 


20.8 


21.5 


22 1 


22.6 


22.7 


23.0 


23.0 


22.8 


22.4 


700 


14.1 


15.2 


16.8 


17.9 


18.8 


20.0 


22 1 


21.5 


22.2 


22.6 


22.9 


23 


23.2 


720 


11.5 


12.7 


13.9 


15.0 


16.4 


17.9 


18.6 


19.7 


20.8 


21.6 


22.3 


22 7 


23.0 


740 


8.9 


9.8 


10.9 


12.2 


13.6 


14.8 


16.2 


17.5 


18.7 


19.5 


20.6 


21.6 


22.3 


760 


5.9 


6.8 


8.0 


9.3 


10.3 


11.8 


13.2 


14.5 


15.9 


17.4 


18.2 


]9.5 


20.5 


780 


4.1 


4.9 


5.6 


6.4 


7.5 


8.6 


9.9 


11.1 


12.6 


14.0 


15.6 


16.8 


18.1 


800 


3.1 


3.3 


4.4 


4.8 


5.5 


6.1 


6.9 


7.9 


9.4 


10.7 


12.1 


13.4 


14.9 


820 


3.1 


3.1 


3.2 


3.1 


3.6 


3.9 


4.8 


5.7 


65 


75 


8.7 


10.0 


11.5 


840 


4.4 


3.7 


3.5 


3.2 


3.2 


3.1 


3.4 


3.7 


4.1 


5.0 


6.2 


7.0 


8.2 


860 i 


6.3 


5.5 


4.6 


4.1 


3.6 


3.4 


3.3 


3.2 


3.4 


3.4 


4.0 


4.5 


5.6 


880 1 


8.6 


7.6 


6.7 


5.9 


5.2 


4.5 


4,1 


3.8 


3.5 


34 


3.4 


3.6 


3.9 


900 


10.9 


10.0 


9.1 


8.3 


7.2 


6.5 


5.8 


5.1 


4.4 


42 


3.8 


3.6 


3.6 


920 


13.1 


12.1 


11.2 


10.3 


9.6 


8.7 


7.7 


6.9 


6.3 


5.8 


5.1 


4.6 


4.2 


940 1 


14.8 


14.1 


13.1 


12.4 


11.5 


10.8 


9.8 


9.1 


8.3 


7.6 


6.8 


6.5 


5.9 


960! 


16.3 


15.4 


14.6 


14.0 


13.2 


12.6 


11.7 


11.0 


10.1 


9.6 


8.8 


8.1 


7.5 


980 


17.7 


16.8 


16.2 


15.2 


14.5 


13.9 


13.1 


12.5 


11.8 


11.2 


10.5 


9.7 


9.3 


1000 


19.3 
480 


18.3 
490 


17.4 
500 


16.6 
510 


15.7 
520 


15.0 
530 


14.2 
540 


13.6 


13.1 
500 


12.3 
570 


11.7 
580 


11.3 
590 


10.8 
600 




550 



TABLE XXX. 



27 



Perturbations produced by Venus, 

Arguments II. and III. 

III. 



. .I. 600 610 620 


630 


640 650 


660 


670 680 690 


700 


710 


720' 


; 7, 7, ~ 


>f 


„ 


~7, 


o : 


9.5 


9.1 


8.4 7.9 


7.4 


7.0 6.6 6.3 


5.9 


5.5 


5.4 


20 11. 5 11.3 


10.7 10.4 


9.S 


9.4 


S.9 


8.5 7.9 7.7 


7.3 


6.7 


6.6 


40 


12.3 12.0 


11.5 


11.0 10.7 


10.3 10.0 


9.6 9.3 


8.9 


8.5 


8.1 


7.3 


60 


13.3 




12.1 


11.6 


11.2 


10.9 10.5 


10.2 10.0 


9.8 


9.5 


9.2 


: s.9 


SO 


14.8 


13.6 


12.9 


12.4 


ll.S 


11.3 


10.9 


10.7 10.3 99 


9.8 


9.S 


9.6 


100 


16.6 


15.4 


14.4 


13 4 


12.6 


12.1 


11.5 


11.0 10.6 10.2 


10.0 


9.9 


9.6 


120 




■ 


16.4 


15.3 


14.3 


13.2 12.4 


II. 6 11.2 10.6 


10.1 


10.1 


9.6 


Uo 


21.1 




13.9 


17.7 


16.5 


15.2 


14.2 


13.0 12.3 


11.6 


11.1 


10.3 


9.9 


160 




22.9 


21.5 


20.4 19.2 


17.9 


16.6 


15.3 14.1 


13.1 


12.0 


11.2 


10.5 


ISO 


3 




23.9 


22.9 21.6 


20.6 


19.1 


18.0 16.7 


15.5 


14.3 


12.9 


12.0 


200 


26.4 


26.0 


25.6 


24.9 24.0 


• 22.9 


21.7 


20.8 19.3 


18.1 


16.9 


15.5 


14.4 


220 






26.1 


25,3 25.3 


24.9 


24.1 


23.1 21.2 


20.9 


19.7 


18.3 


17.1 


210 




25.1 


25.1 


25.3 


25.2 


25.1 


24.7 


24.3 


24.0 


23.0 


21.9 


21.3 


20.2 


2 


23.6 


23.9 


242 


24.5 


:. I i 


24.3 


24.9 




24.3 


23.8 










23.3 


23.6 


23.9 






24.8 


25.0 


24.9 24.9 




24.4 




23.5 


300 




23.0 


23.3 


23.4 


23. S 


24.0 24.1 


24.5 


24.5 


24.6 


24.5 


24.4 


24.0 




22.9 


23.0 


23.1 


23.2 


23.4 


23.3 


23.6 


23 3 


24.0 


23.9 


24.2 


24.2 


24.2 


340 


23.5 


23.5 


23.5 


23.4 


23.5 


23.6 


23.6 


23.5 


23.5 


23.6 


23.9 


23i- 


23.8 


360 






24.3 


242 


24.2 


24.0 


23.7 


23.9 


24.0 


23.7 


23.7 


23.6 


23.6 


; 


24.5 


24.6 




25.1 




24.9 


25.0 


24.9 


24.6 


245 


24.5 


24.3 


24.0 






243 


247 


25.0 


25.4 


25.7 


25.7 


25.5 


25.5 




25.2 


24,8 


24.6 






23.0 


23.7 


24.6 


25.0 


25.7 


26.1 


26.2 


26.3 


26.5 


26.2 


26.0 


25.9 


440 


19.5 


20.3 


21.7 


22 7 


23.7 


246 


25.4 


26.0 


26.5 


26.7 


26.9 


27.0 


26.9 


460 


1 : 


17.3 


19.0 


20.1 


21.4 


22.3 23.5 


24.8 


25.4 


26.1 


26.7 


27.1 


27.3 






14.5 


15.6 


IS. 5 


19.7 


20.9 


22.1 


23.2 


24.4 


25.4 


26.2 


26.8 






12.0 


13.0 


13.S 


14.9 


16.3 


17.9 


19.1 


20.5 


21.6 


22.9 


24.2 


25.1 


520 


9.6 


9.8 


10.5 


11.5 12.4 


13.4 


14.4 


15.5 


17.1 


18.4 


19.9 


21.2 


22.3 


540 


9.3 


9.0 


9.2 


9.6 


10.3 


110 


11.9 


12.8 


13.9 


15.1 


16.5 


17.9 


19.4 


r 


1( J 


1 : 


9.3 


9.1 


9.1 


9.4 


10.0 


10.6 


11.5 


12.4 


13.3 


14 5 


16.0 


580 


12.2 


11.3 


10.4 


9.9 


9.4 


9.0 


9.2 


9.3 


9.7 


10.4 


11.0 


12.0 


12.7 


600 


14.4 


13.3 


12.5 


11.6 


10.3 


10.1 


9.6 


9.4 


9.1 


9.3 


9.9 


10.0 


10.8 


620 


16.9 


16.1 


14.9 


13.7 


12.7 


12.0 


11.1 


10.4 


9.8 


9.5 


9.5 


9.3 


9.7 


640 


19.6 


13.4 


17.4 


16.3 


15.2 


14.2 


13.1 


12.1 


11.3 


10.6 


10.1 


9.6 


9.5 


660 


21.3 


20.6 


19.9 


18.7 


17.8 


16.7 


15.6 


14.4 


13.4 


12.4 


11.7 


11.0 


10.2 




22 4 


22.0 


21.5 


20,3 


20.2 


19.0 


1S.1 


17.0 


15.8 


14.7 


13.7 


12.8 


12.0 


700 


23.2 


,3.2 


22.6 


22 2 


21.7 


21.0 


20.5 


19.3 


18.3 


17.3 


16.0 


15.0 


14.1 




23.0 


233 


23.2 


23.4 


23.1 


22.4 


21.9 


21.3 


20.8 


19.5 


18.5 


17.6 


16.4 






22.S 


23.2 


23.4 23.6 


23.6 


23.3 


22.8 


22.2 21.6 


21.1 


19.9 


18.8 


7 


20.5 


21.4 


22.5 


22.S 


23.3 23.7 


23.6 


23. S 


23.5 23.3 


22.7 




21.3 


7S0 




19.2 


20.4 


21.3 


22.3 


23.0 


23.3 


23.7 


23.S 24.0 


23.8 


23.5 


23.0 


800 


14.9 16.4 


17.7 


19.1 


20.1 


21.2 


21.1 


22.9 


23.4 23.8 


24.1 


24.2 


23.9 


S20 


11.5 12.9 


14.3 


15.3 


17.3 


13.7 


20.0 


20.9 


22.0 22.7 


23.5 


23.9 


24.0 


840 


8.2 9.5 


10.3 12.2 


13,3 


15.2 


16.6 


1S.1 


19.5 20.6 


21.7 


22.6 


23.3 


860 


5.6 6,3 


7.7 


S.8 


10.2 


11.5 


13.2 


14.7 


16.0 17.4 


19.0 


20.2 


21.3 


8S0 


3.9 


4.4 


5.2 


6.1 


7.2 


8.2 


9.7 


10.9 


12.5 14.1 


15.4 


16,3 


18.2 


900 


3.6 


3.6 


3.9 


4.2 


5.0 


5.7 


6.6 


7.8 


9.1 


10.3 


11.8 


13.4 


14.8 


920 


4.2 


3.S 


3.9 


3.9 


4.0 


4.3 


4.7 


5.4 


6.4 


7.3 


8.6 


9.S 




940 


5.9 


5.1 


4.6 


4.4 


4.2 


4.3 


4.3 


4.3 


4.9 


5.3 


6.3 


7.0 


8.0 


960 


7.5 


6.9 


6.3 


5.8 


5.3 


4.7 


4.7 


4.6 


4.6 


4.6 


4.9 


5.4 6.0 


930 


9.3 


3.7 


7.9 


7.4 


6,3 


6.4 


6.0 


5.6 


5.2 


5.0 


4.9 


5.1 5.1 


1000 




10.S 
600 


10.2 
610 


11 

620 


9.1 
630 


8.4 

640 


11 

650 


7.4 
660 


7.0 

670 


6.6 


6.3 


5.9 


5.5 


5.4 
720 


6S0 


690 


700 


710 



28 



TABLE XXX. 



Perturbations produced by Venus, 

Arguments II. and III. 

IIL 



11. 




1 720 
5.4 


730 


1 740 


750 
6.0 


760 
6.3 


770 
6.8 


780 
7.6 


790 
8.4 


800 
9.3 


810 


820 
11.7 


830 
12.9 


840 
14.3 


5.5 


| 5.8 


10.4 


20 


6.6 


6.3 


j 6.0 


6.1 


6.1 


6.2 


6.5 


6.9 


7.7 


8.3 


9.4 


10.2 


11.2 


40 


7.8 


7.4 


1 7.1 


7.0 


6.7 


6.6 


6.8 


6.8 


6.9 


7.2 


7.7 


8.5 


9.3 


60 


8.9 


8.8 


8.3 


8.1 


7.8 


7.6 


7.4 


7.4 


7.3 


7.4 


7.4 


7.7 


8.3 


80 


9.6 


9.5 


i 9.1 


9.1 


9.0 


8.8 


8.4 


8.2 


8.1 


8.1 


8.0 


8.1 


8.2 


100 


9.6 


9.5 


, 96 


9.5 


9.5 


9.3 


9.3 


9.2 


9.2 


9.0 


8.7 


8.7 


8.7 


120 


9.6 


9.6 


! 9.5 


9.3 


9.4 


! 9.6 


9.6 


9.5 


9.5 


9.6 


9.6 


9.6 


9.6 


140 


9.9 


9.5 


9.6 


9.4 


9.3 


i 9.3 


9.0 


9.3 


9.5 


9.8 


9.7 


9.8 


10.0 


160 


10.5 


9.9 


| 9.5 


9.1 


8.9 


9.0 


8.9 


9.0 


9.0 


9.0 


9.5 


9.6 


9.9 


ISO 


12.0 


11.0 


10.1 


9.7 


9.1 


8.8 


8.7 


8.3 


8.5 


8.7 


8.8 


9.0 


9.1 


200 


14.4 


13.3 


12.0 


11.0 


10.1 


9.4 


8.9 


8.5 


8.2 


8.0 


8.0 


8.3 


8.5 


220 


17.1 


15.7 


14.6 


13.2 


12.0 


10.9 


10.2 


9.2 


8.7 


8.3 


7.9 


7.7 


7.7 


240 


20.2 


19.1 


17.8 


16.5 


14.5 


13.4 


12.2 


11.1 


10.0 


9.4 


8.4 


8.0 


7.7 


260 21.6 


21.1 


20.1 


19.2 


17.3 


15.9 


14.6 


13.4 


12.4 


11.3 


10.1 


9.1 


8.6 


280 23.5 


22.7 


21.6 


21.0 


19.8 


18.8 


17.3 


16.1 


15.0 


13.5 


12.5 


11.5 


10.2 


300 24.0 


23.4 


23.2 


22.4 


21.4 


20.5 


19.8 


1S.7 


17.5 


16.1 


15.0 


13.7 


12.4 


320 ' 24.2 


23.9 


23.5 


23.1 


22.7 


22.2 


21.2 


20.6 


19.6 


18.6 


17.5 


16.3 


15.1 


340 23.8 


23.9 23.7 


23.5 


23.2 


22.8 


22.3 


21.4 


20.9 


20.5 


19.2 


18.6 


17.4 


360 23.6 


23.6 23.6 


23.3 


23.3 


23.1 


22.9 


22.4 


22.0 


21.4 


20.4 


19.9 


1S.9 


380 


24.0 


24.0 23.7 


23.5 


23.3 


23.1 


23.1 


22.7 


22.4 


22.2 


21.6 


20.8 


20.0 


400 


24.6 


24.4 24.4 


24.0 


23.8 


23.4 


23.2 


23.0 


22.8 


22.4 


22.1 


21.6 


21.3 


420 


25.9 


25.6 25.2 


24.S 


24.7 


24.3 


23.9 


23.6 


23.3 


22.9 


22.7 


22.3 


21.7 


440 


26.9 


26.6 26.4 


20.2 


25.9 


25.5 


25.2 


24.9 


24.5 


23.8 


23.4 


23.0 


22.8 


460 


27.3 


27.6 27.6 


27.4 


27.0 


26.9 


26.5 


28.1 


25.6 


25.0 


24.6 


24.2 


23.7 


480 


26.8 


27.4 27.6 


28.0 


2S.1 


28.2 


27.7 


27.4 


27.3 


26.6 


26.2 


25.7 


25.1 


500 


25.1 


26.1 26.8 


27.5 


28.1 


28.2 


28.6 


2S.5 


28.4 


28.3 


27.6 


27.2 


26.7 


520 


22.3 


23.9 24.8 


25.9 


26.8 


27.5 


28.1 


28.5 


28.7 


29.0 


28.8 


28.6 


28.4 


540 


19.4 


20.7 22.1 


23.4 


24.6 


25.6 


26.5 


27.4 


28.0 


28.7 


28.9 


29.1 
28.6 


29.2 


560 


16.0 


17.3 18.6 


19.9 


21.4 


22.9 


24.1 


25.5 


26.4 


27.3 


28.2 


29.2 


580 


12.7 


14.1 15.5 


16.8 


18.0 


19.3 


20.9 


22.2 


23.5 


24.9 


26.1 


27.0 


27.8 


600 


10.8 


11.6 12.7 


13.6 


14.9 


16.2 


17.5 


18.7 


20.2 


21.8 


23.0 


24.4 


25.5 


620 


9.7 


10.0 10.5 


10.7 


12.2 


13.2 


14.4 


15.6 


17.0 


18.3 


19.6 


21.2 


22.6 


640 


9.5 


9.4 9.6 


10.1 


10.4 


11.1 


12.0 


13.0 


14.0 


15.2 


16.5 


17.9 


19.2 


660 


10.2 


10.0 9.7 


9.5 


9.5 


9.9 


10.4 


11.0 


11.7 


12.7 


13.8 


14.9 


16.2 


680 


12.0 


11.2 10.5 


10.0 


9.7 


9.5 


9.6 


10.0 


10.4 


11.0 


11.6 


12.5 


13.8 


700 


14.1 


13.1 i 12.3 


11.3 


10.7 


10.1 


9.7 


9.7 


9.9 


9.9 


10.4 


10.9 


11.5 


720 


16.4 


15.3 


14.4 


13.3 


12.2 


11.6 


10.9 


10.2 


10.1 


9.9 


10.0 


10.1 


10.4 


740 


18.8 


17.7 


16.7 


15.6 


14.4 


13.5 


12.4 


11.5 


11.1 


10.7 


10.1 


10.0 


10.3 


760 


21.3 


20.1 


19.2 


18.1 


16.6 


15.6 


14.7 


13.6 


12.8 


11.9 


11.3 


10.7 


10.3 


780 


23.0 


22.3, 


21.5 


20.5 


19.4 


18.4 


17.2 


15.8 


14.9 


14.0 


13.0 


12.2 


11.3 


800 


23.9 


23.9 


23.4 


22.6 


21.9 


20.7 


19.8 


18.8 


17.5 


16.2 


15.1 


14.2 


13.4 


820 


24.0 


24.5, 


24.2 


23.9 


23.3 


22.6 


22.3 


21.3 


20.3 


19.4 


18.3 


17.3 


16.2 


840 


23.3 


24.0 i 


24.3 


24.5 


24.4 


24.3 


23.8 


23.4 


22.7 


21.7 


20.8 


19.6 


18.3 


860 


21.3 


22.3, 


23.3 


23.9 


24.2 


24.7 


24.5 


24.5 


24.3 


23.6 


23.1 


21.9 


21.0 


880 


18.2 


19.7, 


20.9 


22.0 


22.8 


23.8 


24.1 


24.6 


24.8 


24.7 


24.5 


24.0 


23.5 


900 


14.8 


16.1 


17.6 


19.0 


20.6 


21.5 


22.5 


23.2 


24.1 


24.5 


24.2 


24.8 


24.5 


920 


11.2 


12.6; 


14.0 


15.5 


17.0 


18.4 


19.9 


21.0 


22.0 


22.9 


23.5 


24.5 


24.5 


940 


8.0 


9.3 i 


10.7 


12.0 


13.3 


14 8 


16.4 


17.6 


19.1 


20.4 


21.4 


22.4 


23.2 


960 


6.0 


6.9 | 


7.8 


8.6 


10.2 


11.5 


12.7 


14.1 


15.6 


16.9 


18.5 


19.5 


20.7 


980 


5.1 


5.5 j 


6.0 


6.7 


7.7 


8.5 


S.7 


10.9 


12.2 


13.6 


14.8 


16.1 


17.6 


1000 


5.4 
720 


5.5 


5.8 


5.8 
750 


6.3 


6.8 
770 


7.6 

780 


8.4 
790 


9.3 

800 


10.5 
810 


11.7 
820 


12.9 
830 


14.3 
840 

—4 


730 | 


740 


760 



TABLE XXX. 

Perturbations produced by Venus. 

Arguments II. and III. 

III. 



II. " 




840 
14.3 


850 
15.5 


860 


870 


r 880 


890 
20.2 


900 


910 


920 


930 


940 


950 


960 


16.9 


18.2 


19.2 


21.4 


22.5 


23.0 


23.5 


24.0 


24.2 


24.2 


20 


11.2 


12.4 


13.6 


14.9 


16.2 


17.3 


18.6 


19.6 


20.5 


21.5 


22.4 


23.1 


23.6 


40 


9.3 


10.2 


10.9 


11.8 


13.3 


14.2 


15.5 


16.6 


17.8 


18.8 


19.7 


20.7 


21.6 


60 


S.3 


8.7 


9.5 


10.1 


10.8 


11.6 


12.7 


13.8 


14.9 


15.9 


17.0 


18.1 


19.1 


SO 


8.2 


8.3 


8.6 


8.9 


9.6 


10.3 


10.7 


11.6 


12.5 


13.3 


14.5 


15.2 


16.2 


100 


S.7 


8.7 


8.9 


9.0 


9.1 


9.4 


9.9 


10.4 


11.0 


11.7 


12.4 


12.9 


14.0 


120 


9.6 


9.5 


9.3 


9.6 


9.6 


9.7 


9.9 


9.8 


10.4 


10.9 


11.3 


11.8 


12.3 


140 


10.0 


10.2 


10.1 


10.2 


10.1 


10.3 


10.4 


10.5 


10.5 


10.6 


10.9 


11.4 


11.5 


160 


9.9 


10.0 


10.2 


10.4 


10.6 


11.0 


11.0 


10.9 


11.0 


11.3 


11.3 


11.3 


11.6 


180 


9.1 


9.6 


9.9 


10.1 


10.4 


10.7 


11.0 


11.3 


11.5 


11.7 


11.7 


11.9 


12.2 


200 


8.5 


8.8 


9.1 


9.5 


9.7 


10.0 


10.5 


11.0 


11.2 


11.6 


12.0 


12.2 


12.4 


220 


7.7 


7.7 


8.1 


8.4 


8.8 


9.2 


9.7 


10.1 


10.6 


11.0 


11.4 


11.8 


12.3 


240 


7.7 


7.3 


7.4 


7.4 


7.7 


8.0 


8.4 


9.0 


9.6 


10.0 


10.5 


11.0 


11.5 


260 


8.6 


7.9 


7.4 


7.2 


7.1 


7.1 


7.3 


7.6 


8.1 


8.5 


9.3 


10.0 


10.4 


280 


10.2 


9.2 


8.3 


7.9 


7.4 


7.1 


7.0 


6.9 


7.0 


7.3 


7.7 


8.5 


8.8 


300 


12.4 


11.4 


10.4 


9.3 


8.5 


7.8 


7.4 


6.9 


6.7 


6.8 


6.8 


7.0 


7.5 


320 


15.1 


13.9 


12.5 


11.4 


10.5 


9.7 


8.6 


7.8 


7.4 


7.0 


6.6 


6.5 


6.7 


340 


17.4 


16.4 


15.2 


13.9 


12.7 


11.6 


10.6 


9.7 


8.7 


8.0 


7.3 


6.8 


6.6 


360 


18.9 


18.1 


17 4 


16.3 


15.1 


13.8 


12.8 


11.7 


10.6 


9.8 


8.8 


8.0 


7.4 


380 


20.0 


19.6 


18.8 


17.7 


16.9 


15.0 


15.1 


13.9 


12.7 


11.8 


10.8 


9.8 


8.9 


400 


21.3 


20.6 


19.6 


19.4 


18.4 


17.6 


16.5 


15.7 


14.8 


13.7 


12.8 


11.8 


10.9 


420 


21.7 


21.1 


20.8 


20.3 


19.3 


18.9 


18.2 


17.2 


16.3 


15.3 


14.5 


13.7 


12.6 


440 


22.8 


22.1 


21.6 


20.8 


20.6 


19.7 


19.0 


18.6 


17.7 


16.6 


15.9 


15.1 


14.2 


460 


23.7 


23.3 


22.7 


22.0 


21.6 


20.9 


20.2 


19.5 


18.5 


18.1 


17.3 


16.7 


15.7 


180 


25.1 


24.4 


23.9 


23.3 


22.8 


22.0 


21.4 


20.9 


20.2 


19.3 


18.3 


17.7 


16.9 


500 


26.7 


26.3 


25.7 


24.9 


24.3 


23.6 


23.0 


22.3 


21.4 


20.7 


20.3 


19.1 


18.1 


520 


28.4 


27.8 


27.3 


26.8 


26.3 


25.6 


24.7 


23.9 


23.3 


22.6 


21.8 


20.8 


20.1 


540 


29.2 


29.2 


28.9 


28.5 


27.8 


27.4 


26.8 


26.1 


25.3 


24.4 


23.7 


23.0 


22.0 


560 


29.2 


♦29.3 


29.5 


29.6 


29.3 


29.1 


28.8 


28.0 


27.4 


26.9 


26.1 


25.1 


24.3 


580 


27.8 


28.6 


29.0 


29.4 


29.6 


29.8 


29.8 


29.3 


28.0 


28.7 


27.9 


27.3 


26.6 


600 


25.5 


26.7 


27.6 


28.4 


28.9 


29.2 


29.6 


29.9 


29.9 


29.8 


29.3 


29.0 


28.5 


620 


22.6 


23.8 


25.0 


26.2 


27.1 


27.9 


28.8 


29.3 


29.6 


29.8 


30.1 


29.8 


29.6 


640 


19.2 


20.6 


21.6 


23.3 


24.6 


25.2 


26.6 


27.8 


28.3 


28.9 


29.4 


29.7 


29.9 


660 


16.2 


17.5 


18.8 


20.2 


21.1 


22.9 


24.0 


25.1 


26.2 


27.1 


28.2 


28.8 


29.2 


680 


13.8 


14.7 


15.8 


16.9 


18.4 


19.9 


20.6 


22.3 


23.6 


24.9 


25.8 


26.7 


27.5 


700 


11.5 


12.3 


13.4 


14.6 


15.6 


16.7 


18.0 


19.5 


20.7 


22.0 


23.1 


24.2 


25.1 


720 


10.4 11.0 


11.4 


12.3 


13.3 


14.3 


15.6 


16.4 


17.7 


19.3 


19.9 


21.6 


22.6 


740 


10.3 


10.4 


10.5 


11.0 


11.4 


12.2 


13.3 


14.2 


15.3 


16.5 


17.4 


18.8 


19.5 


760 


10.3 


10.0 


10.2 


10.3 


10.7 


11.0 


11.5 


12.2 


13.1 


14.2 


15.1 


16.0 


17.3 


780 


11.3 


10.8 


10.6 


10.2 


10.2 


10.5 


10.7 


11.1 


11.5 


12.3 


13.2 


14.0 


15.0 


800 


13.4 


12.5 


11.7 


11.0 


10.6 


10.3 


10.3 


10.4 


10.7 


11.0 


11.6 


11.3 


12.2 


820 


16.2 


15.2 


14.4 


13.5 


13.5 


11.9 


11.4 


11.0 


10.9 


10.8 


10.8 


11.2 


11.4 


840 


18.3 


17.1 


16.2 


14.9 


14.1 


13.0 


12.4 


11.7 


11.2 


10.7 


10.6 


11.1 


11.2 


860 


21.0 


20.2 


18.7 


17.7 


16.6 


15.4 


14.3 


13.3 


12.5 


11.9 


11.4 


11.0 


10.9 


880 


23.5 


22.4 


21.3 


20.4 


19.3 


18.0 


17.0 


15.9 


14.8 


13.7 


12.8 


12:0 


12.6 


900 


24.5 


24.2 


23.8 


22.7 


21.9 


19.9 


19.7 


18.6 


17.2 


16.4 


15.3 


14.1 


13.3 


920 


24.5 


24.8 


24.7 


243 


24.1 


23.2 


22.3 


21.3 


20.0 


19.3 


18.0 


16.7 


15.7 


940 


23.2 


24.0 


24.5 


24.6 


24.5 


24.5 


24.2 


23.5 


22.7 


21.8 


20.6 


19.5 


18.4 


960 


20.7 


21.9 


22.8 


23.6 


24.0 


24.5 


24.5 


24.2 


24.3 


23.7 


22.9 


22.1 


21.0 


980 


17.6 


18.7 


20.1 


21.2 


22.2 


23.1 


23.6 


24.0 


24.3 


24.3 


24.3 


23.7 


23.0 


1000 


14.3 


15.5 


16.9 


18.2 


19.2 


20.2 


21.4 
900 


22.5 


23.0 


23.5 


24.0 


24.2 


24.2 




840 


850 


860 


870 


880 


890 


910 


920 


930 


940 


950 


960 



30 



TABLE XXX. XXXI. 



Perturbations by Venus. 

Arguments II and III. 

III. 



Perturbations by Mars. 

Arguments II and IV. 

IV. 



[L 


960 


970 


980 


990 , 1000 

! 


| 10 , 


20 | 30 


10 


50 


60 


70. 




~^ 


~ 


~^~ 


„ 


" 


~7~ 


// 


~ 


" 


__ 


~7T~ 


„ 


,r \ 





24.2 


23.7 


23.1 


22.5 


21.6 


9.5 


10.2 


10.8 


11.2 


11.5 


11.7 


11.8 


11.5 


20 


23.6 


23.7 


24.0 


23.4 


23.1 


8.3 


9.1 


9.8 


10.5 


10.9 


11.2 


11.5 


11.6 


40 


21.6 


22.4 


22 9 


23.5 


23.5 


7.1 


7.9 


8.S 


9.4 


10.0 


10.6 


10.8 


11.2 


60 


19.1 


20.1 


20.7 


21.5 


22.2 


5.8 


6.7 


7.6 


8.4 


9.1 


9.8 


10.3 


10.5 


80 


16.2 


17.3 


13.4 


19:7 


20.0 


4.3 


5.3 


6.4 


7.2 


8.0 


8.9 


9.3 


9.9 


100 


14,0 


14.8 


15.6 


16.5 


17.6 


3.3 


4.2 


5.0 


5.9 


6.8 


7.6 


8.4 


9.1 


120 


12.3 


12.9 


13.7 


14.3 


15.3 


2.4 


3.1 


3.9 


4.8 


5.6 


6.4 


7.3 


8.0 


140 


11.5 


12.0 


12.6 


12.8 


13.6 


2.1 


2.4 


2.9 


3.8 


4.6 


5.5 


6.3 


7.0 


160 


11.6 


11.8 


12.1 


12.3 


12.7 


2.0 


2.2 


2.4 


2.7 


3.5 


4.4 


5.1 


5.9 


180 


12.2 


12.2 


12.3 


12.5 


12.7 


1.9 


2.0 


2.3 


2.6 


2.9 


3.4 


3.9 


4.9 ' 


200 


12.4 


12.7 


12.8 


13.1 


13.2; 


2.3 


2.2 


2.2 


2.4 


2.7 


3.0 


3.4 


3.8 


220 


12.3 


12.7 


13.0 


13.3 


13.5 


3.0 


2.0 


2.5 


2.4 


2.5 


2.7 


3.1 


3.5 


240 


11.5 


12.1 


12.4 


13.1 


13.6 


3.7 


3.3 


3.0 


2.9 


2.7 


2.8 


2.9 


3.2 


260 


10.4 


11.0 


11.5 


12.2 


12.8 


4.8 


4.1 


3.7 


3.5 


3.1 


3.1 


3.0 


3.1 


280 


8.8 


9.6 


10.4 


10.7 


11.5 


5.5 


5.1 


4.6 


4.1 


3.8 


3.5 


3.5 


3.4 


300 


7.5 


7.9 


8.6 


9.0 


io.i : 


6.2 


5.8 


5.6 


5.0 


4.8 


4.2 


3.9 


3.8 


320 


6.7 


6.S 


7.3 


7.8 


8.3 ' 


6.9 


6.6 


6.1 


5.9 


5.4 


5.1 


4.7 


4.3 


310 


6.6 


6.4 


6.6 


6.7 


6.2 


7.2 


7.1 


6.9 


6.5 


6.2 


5.8 


5.5 


5.1 


360 


7.4 


6.9 


6.5 


6.5 


6.5 


7.5 


7.4 


7.1 


7.0 


6.8 


6.4 


6.2 


5.8 


3S0 


8.9 


8.2 


7.5 


6.9 


6.8 


7.5 


7.6 


7.3 


7.3 


7.2 


7.1 


6.7 


6.5 


400 


10.9 


10.0 


9.0 


8.3 


7.5 


7.3 


7.3 


7.5 


7.4 


7.4 


7.4 


7.1 


7.0 


420 


12.6 


11.6 


10.7 


9.9 


9.1 


6.9 


7.0 


7.3 


7.4 


7.4 


7.4 


7.3 


7.5 


410 


14.2 


13.3 


12.5 


11.6 


10.6 


6.5 


6.8 


6.8 


7.1 


7.2 


7.3 


7.3 


7.4 


460 


15.7 


14.8 


13.9 


13.0 


12.1 


6.2 


6.2 


6.5 


6.7 


6.8 


7.1 


7.1 


7.3 


430 


16.9 


16.3 


15.5 


14.5 


13.6 


5.8 


5.9 


6.0 


6.2 


6.4 


6.5 


7.0 


6.9 


500 


1S.1 


17.6 


16.6 


15.8 


15.1 


5.3 


5.4 


5.7 


5.8 


6.0 


6.0 


6.3 


6.6 


520 


20.1 


19.2 


18.1 


17.4 


16.5 


5.1 


5.1 


5.1 


5.3 


5.4 


5.6 


5.8 


6.0 


540 


22.0 


21.0 


20.2 


19.2 


18.1 


4.7 


4.8 


4.8 


4.8 


5.0 


5.1 


5.4 


5.5 


560 


24.3 


23.5 


22.6 


21.5 


20.6 


4.4 


4.5 


4.6 


4.6 


4.7 


4.8 


4.8 


5.0 


5S0 


26.6 


25.7 


24.9 


23.8 


23.0 


4.2 


4.3 


4,4 


4.3 


4.5 


4.4 


4.4 


4.5 


600 


28.5 


27.8 


27.0 


26.3 


25.4 


4.0 


4.2 


4.3 


4.2 


4.2 


4.2 


4.2 


4.3 


620 


29.6 


29.2 


28.8 


28.2 


27.4 


4.2 


4.0 


4.1 


4.0 


4.0 


4.0 


4,0 


3.9 


640 


29.9 


30.0 


29.9 


29.5 


29.5 


4.3 


4.2 


4.1 


4.0 


4.1 


4.0 


3.9 


3.9 


660 


29.2 


29.5 


29.7 


29.8 


29.9 


4.6 


4.4 


4,3 


4.1 


4.1 


4.1 


4.0 


3.8 


6S0 


27.5 


28.6 


28.9 


29.2 


29.7 


- 4.8 


4.6 


4.5 


4.3 


4.2 


4.1 


4.0 


3.9 


700 


25.1 


26.4 


27.3 


27.8 


28.7 


5.3 


5.0 


4.8 


4.5 


4.6 


4.0 


4.1 


4.1 


720 


22.6 


23.9 


25.0 


26.1 


26.8 


5.8 


5.5 


5.1 


5.0 


4.7 


4.5 


4.1 


4.1 


740 


19.5 


21.3 


22.5 


23.6 


24,6 


6.5 


6.1 


5.7 


5.4 


5.2 


4.9 


4.6 


4.3 


760 


17.3 


18.6 


19.4 


21.0 


22.1 


7.4 


6.7 


6.4 


6.0 


5.6 


5.3 


5.1 


5.0 


780 


15.0 


15.8 


17.1 


18.5 


19.3 


8.2 


7.6 


6.9 


6.5 


6.4 


5.8 


5.6 


5.3 


800 


12.2 


14.1 


14.8 


15.9 


17.0 


9.2 


8.5 


8.0 


7.3 


6.8 


6.5 


6.1 


5.8 


820 


11.4 


12.0 


12.5 


13.4 


15.4 


10.1 


9.6 


8.8 


8.2 


7.6 


7.1 


6.7 


6.5 


840 


11.2 


11.3 


11.7 


12.2 


13.2 


10.9 


10.4 


9.8 


9.1 


8.4 


7.9 


7.5 


6.9 


860 


10.9 


10.8 


10.9 


11.2 


11.5 


11.7 


11.0 


10.4 


10.0 


9.4 


8.7 


8.2 


7.7 


8S0 


12.6 


11.3 


11.1 


10.8 


11.0 


12.3 


11.9 


11.3 10.6 


10.2 


9.7 


8.9 


8.4 


900 


13.3 


12.3 


12.9 


11.3 


11.2 


12.4 


12.2 


11.8 11.6 


10.8 


10.3 


9.7 


9.3 


920 


15.7 


14.6 


13.7 


12.8 


12.1 


12.3 


12.3 


12.2* 11.9 


11.6 


11.0 


10.5 


9.9 


940 


18.4 


17.3 


16.2 


14.5 


14.0 


12.1 


12.1 


12.2 12.2 


11.8 


11.4 


11.0 


10.6 


960 


21.0 


20.0 


18 9 


17.9 


16.7 


11.4 


11.9 


11.9 12.0 


12.0 


11.7 


11.4 


11.0 


980 


23.0 


22.4 


21.4 


20.3 


19.5 


10.6 


11.1 


11.6, 11.8 


11.9 


11.9 


11.7 


11.4 


1000 


24.2 


23.7 


23.1 


22.5 


21.6 


9.5 


10.2 


10.8 1 11.2 


11.5 


11.7 


11.8 


11.5 


~~ 


960 


970 


980 


990 


1000 





10 


20 | 30 


40 


50 


60 


70 



TABLE XXXI. 



31 



Perturbations produced by Mars 

Arguments II and IV. 

IV. 



/ 



II. 


70 


80 


90 


100 


110 


120 


130 


140 


150 


160 




170 


180 


190 


200 






„ 


//• 


,, 


fr 


„ 


/.- 




~~z 


>> 


/, 


~77~ 




T 





11.5 


11.2 


11.0 


10.6 


10.1 


9.9 


9.5 


9.0 


8.6 


8.2 


8.1 


7.8 


7.6 


7.4 


20 


11.6 


11.4 


11.0 


10.9 


10.6 


10.2 


9.7 


9.1 


9.1 


8.8 


8.4 


8.1 


7.9 


7.8 


40 


11.2 


11.3 


11.2 


11.0 


10.8 


10.5 


10.3 


9.8 


9.4 


9.3 


9.1 


8.7 


S.4 


8.2 


60 


10.5 


10.9 


11.1 


10.9 


11.0 


10.9 


10.4 


10.0 


9.7 


9.5 


9.2 


8.8 


S.7 


8.4 


80 


9.9 


10.0 


10.5 


10.9 


10.8 


10.7 


10.4 


10.3 


10.0 


9.7 


9.3 


9.0 


S.8 


8.6 


100 


9.1 


9.5 


9.8 


10.1 


10.6 


10.5 


10.4 


10.3 


10.1 


9.9 


9.6 


9.3 


9.0 


8.8 


120 


8.0 


8.8 


9.3 


9.5 


9.9 


10.2 


10.2 


10.1 


10.0 


9.8 


9.6 


9.4 


9.1 


8.9 


140 


7.0 


7.9 


8.4 


9.0 


9.3 


9.6 


9.9 


9.9 


9.9 


9.7 


9.7 


9.4 


9.3 


8.9 


160 


5.9 


6.5 


7.2 


8.0 


8.5 


8.9 


9.2 


9.6 


9.5 


9.6 


9.5 


9.5 


9.3 


9.1 


180 


4.9 


5.6 


6.4 


6.9 


7.7 


8.3 


8.6 


8.9 


9.4 


9.3 


9.3 


9.3 


9.2 


9.1 


200 


3.8 


4.6 


5.3 


6.0 


6.7 


7.4 


7.9 


8.3 


8.0 


8.9 


9.1 


9.0 


9.0 


8.9,' 


220 


3.5 


3.9 


4.4 


5.1 


5.8 


6.4 


7.1 


7.6 


7.9 


8.4 


8.6 


8.8 


8.8 


8.7 


240 


3.2 


3.6 


4.0 


4.4 


5.0 


5.5 


6.2 


6.8 


7.4 


7.6 


8.1 


8.4 


8.4 


8.5 


260 


3.1 


3.2 


3.8 


4.1 


4.5 


4.9 


5.4 


5.9 


6.6 


7.1 


7.5 


7.7 


S.O 


8.2 


280 


3.4 


3.4 


3.5 


3.8 


4.2 


4.5 


4.9 


5.5 


5.6 


6.2 


6.8 


7.1 


7.5 


7.8 


300 


3.8 


3.7 


3.7 


3.7 


3.9 


4.4 


4.7 


4.9 


5.4 


5.7 


6.0 


6.6 


6.9 


7.3 


320 


4.3 


4.2 


4.1 


4.0 


4.1 


4.2 


4.4 


4.7 


50 


5.4 


5.8 


6.0 


6.4 


6.6 


340 


5.1 


4.9 


4.6 


4.4 


4,4 


4.3 


4.5 


4.5 


5.0 


5.2 


5.5 


5.8* 


6.0 


6.3 


360 


5.8 


5.6 


5.3 


5.0 


4.8 


4.8 


4.7 


4.8 


4.9 


5.1 


5.4 


5.5 


5.9 


6.1 


380 


6.5 


6.4 


5.9 


5.7 


5.5 


5.4 


5.1 


5.1 


5,1 


5.1 


5.4 


5.5 


5.7 


5.8 


400 


7.0 


6.7 


6.7 


6.3 


6.1 


5.9 


5.7 


5.6 


5.5 


5.5 


5.5 


5.6 


5.7 


5.9 


420 


7.4 


7.2 


6.9 


7.1 


6.7 


6.4 


6.3 


6.1 


6.0 


5.9 


5.9 


5.8 


5.8 


6.1 


440 


7.5 


7.4 


7.4 


7.0 


7.1 


7.4 


6.8 


6.7 


6.5 


6.3 


6.3 


6.4 


6.2 


6.3 


460 


7.3 


7.4 


7.4 


7.5 


7.4 


7.3 


7.3 


7.2 


7.1 


7.1 


6.7 


6.7 


6.7 


6.7 


480 


6.9 


7.1 


7.3 


7.4 


7.5 


7.3 


7.6 


7.5 


7.4 


7.5 


7.4 


7.2 


7.1 


7.1 


500 


6.G 


6.8 


6.9 


7.2 


7.3 


7.5 


7.5 


7.6 


7.8 


7.7 


7.8 


7.7 


7.6 


7.4 


520 


6.0 


6.3 


6.5 


6.7 


7.1 


7.2 


7.5 


7.5 


7.7 


7.8 


7.9 


7.6 


7.9 


7.9 


540 


5.5 


5.7 


6.0 


6.3 


6.6 


6.9 


7.1 


7.3 


7.4 


7.7 


7.9 


8.0 


8.2 


8.3 


560 


5.0 


5.2 


5.4 


5.8 


5.9 


6.2 


6.6 


6.9 


7.1 


7.4 


7.7 


7.8 


8.1 


8.2 


580 


4,5 


4.7 


4,9 


5.0 


5.3 


5.7 


6.0 


6.6 


6.8 


7.1 


7.2 


7.5 


7.9 


8.2 


600 


4.3 


4.3 


4,4 


4.6 


4,6 


5.0 


5.3 


5.6 


5.9 


6.5 


6.9 


7.0 


7.4 


7.7 


620 


3.9 


4.0 


4.0 


4.1 


4.3 


4.4 


4.6 


4.9 


5.3 


5.4 


6.1 


'6.6 


6.9 


7.4 


640 


3.9 


3.8 


3.8 


3.8 


3.9 


3.9 


4.1 


4,3 


4.5 


5.0 


5.2 


5.8 


6.3 


6.7 


660 


3.8 


3.7 


3.7 


3.6 


3.6 


3.7 


3.8 


3.9 


4.1 


4.2 


4.5 


5.0 


5.3 


6.0 


680 


3.9 


3.8 


3.6 


3.4 


3.5 


3.4 


3.5 


3.5 


3.6 


3.7 


3.8 


4.2 


4.6 


4.9 


700 


4.1 


3.9 


3-.S 


3.6 


3.5 


3.3 


3.3 


3.2 


3.2 


3.2 


3.5 


3.6 


3.8 


4.2 


720 


4.1 


4.1 


4.0 


3.8 


3.6 


3.5 


3.3 


3.2 


3.3 


3.2 


3.0 


3.2 


3.4 


3.6 


740 


4.3 


4.3 


4.2 


4.0 


3.8 


3.7 


3.5 


3.2 


3.0 


3.0 


2.9 


2.8 


2.9 


3.1 


760 


5.0 


4.7 


4.4 


4.3 


4.1 


3.8 


3.7 


3.4 


3.1 


3.0 


2.9 


2.7 


2.7 


2.8 


780 


5.3 


5.1 


4.7 


4.6 


4.4 


4.4 


4.0 


3.8 


3.4 


3.2 


2.9 


2.8 


2.7 


2.5 


800 


5.8 


5.5 


5.4 


4.8 


4.7 


4.7 


4.5 


4.2 


3.9 


3.5 


3.3 


2.9 


2.8 


2.7 


820 


6.5 


6.1 


5.8 


5.6 


5.0 


5.0 


4.9 


4.6 


4.3 


4.1 


3.6 


3.3 


3.0 


2.9 


840 


6.9 


6.7 


6.3 


6.1 


5.8 


5.3 


5.2 


4.9 


4.9 


4.5 


4.2 


3.9 


3.5 


3.1 


860 


7.7 


7.4 


6.9 


6.6 


6.2 


6.2 


5.5 


5.4 


5.2 


5.0 


4.8 


4.4 


4.1 


3.6 


880 


8.4 


7.9 


7.6 


7.1 


6.9 


6.4 


6.4 


5.8 


5.7 


5.4 


5.2 


5.0 


4.6 


4.3 


900 


9.3 


' 8.7 


8.3 


7.7 


7.4 


7.1 


6.7 


6.6 


6.1 


6.0 


5.6 


5.4 


5.2 


4.9 


920 


9.9 


9.3 


8.8 


8.4 


7.9 


7.7 


7.3 


6.9 


6.6 


6.3 


6.2 


6.1 


5.6 


54 


940 


10.6 


10.1 


9.5 


8.9 


8.7 


8.2 


7.8 


7.6 


7.2 


7.1 


6.5 


6.5 


6.3 


5.9 


960 


11.0 


10.7 


10.3 


9.7 


9.1 


8.7 


8.4 


8.0 


7.8 


7.4 


7.2 


6.9 


6.7 


6.5 


980 


11.4 


11.0 


10.6 


10.2 


9.8 


9.2 


8.9 


8.4 


8.1 


8.0 


7.6 


7.3 


7.2 


6.9 


1000 


11.5 
70 


112 


11.0 


10.6 


10.0 


9.9 


9.5 


9.0 


8.6 


8.2 


8.1 
170 


7.4 


7.6 


7.4 




80 


90 


100 


110 


120 


130 


1 140 


150 


160 


180 


190 


200 



32 



TABLE XXXI. 



Perturbations produced by Mars. 

Arguments II. and IV. 

IV. 



II, 


200 

7.4 


210 

7.2 


220 
7.0 


230 

6.6 


240 
6.4 


250 


260 


270 


280 


290 


300 


310 


320 





6.2 


5.7 


5.3 


4.9 


4.7 


4.1 


3.8 


3.4 


20 


7.8 


7.2 


7.3 


7.2 


7.0 


6.6 


6.3 


6.0 


5.7 


5.3 


5.0 


4.4 


3.9 


40 


8.2 


8.1 


7.6 


7.5 


7.3 


7.2 


6.8 


6.6 


6.2 


5.9 


5.6 


5.2 


4.7 


60 


8.4 


8.0 


7.9 


7.S 


7.6 


7.5 


7.3 


7.1 


6.8 


6.4 


6.1 


5.8 


5.4 


80 


8.6 


8.5 


8.2 


8.0 


7.6 


7.7 


7.6 


7.4 


7.1 


7.0 


6.7 


6.3 


6.0 


100 


8.8 


8.5 


8.6 


8.4 


8.2 


7.6 


7.7 


7.8 


7.6 


7.3 


7.2 


6.9 


6.6 


120 


S.9 


8.7 


8.4 


8.4 


8.3 


8.3 


8.0 


7.9 


7.7 


7.6 


7.5 


7.3 


7.0 


140 


8.9 


8.7 


8.4 


8.3 


8.2 


8.1 


8.3 


8.0 


7.9 


7.8 


7.7 


7.5 


7.4 


160 


9.1 


8.9 


8.7 


8.4 


8.3 


8.3 


8.2 


8.1 


8.0 


7.9 


7.9 


7.7 


7.6 


180 


9.1 


8.8 


8.7 


8.5 


S.4 


8.2 


8.0 


8.0 


8.1 


7.9 


7.8 


8.0 


7.8 


200 


8.9 


8.8 


8.6 


8.4 


8.4 


8.3 


8.1 


8.0 


7.9 


7.8 


7.8 


7.9 


7.9 


220 


8.7 


S.7 


8.6 


8.4 


8.2 


S.l 


8.0 


7.9 


7.8 


7.7 


7.7 


7.6 


7.7 


240 


8.5 


8.4 


8.5 


8.3 


8.1 


8.0 


7.8 


7.8 


7.8 


7.8 


7.8 


7.8 


7.6 


260 


8.2 


8.2 


8.1 


8.1 


8.1 


7.8 


7.8 


7.7 


7.6 


7.6 


7.6 


7.5 


7.4 


280 


7.8 


7.8 


8.0 


7.8 


7.9 


7.9 


7.7 


7.5 


7.5 


7.3 


7.3 


7.4 


7.3 


300 


7.3 


7.6 


7.5 


7.6 


7.7 


7.6 


7.6 


7.6 


7.4 


7.3 


7.1 


7.0 


7.1 


320 


6.6 


7.1 


7.3 


7.4 


7.4 


7.3 


7.4 


7.4 


7.3 


7.1 


7.0 


7.0 


6.8 


340 


6.3 


6.4 


6.7 


7.2 


7.1 


7.2 


7.2 


7.1 


7.1 


7.0 


6.9 


6.8 


6.8 


360 


6.1 


6.2 


6.4 


6.5 


6.9 


6.9 


7.0 


7.0 


6.9 


6.8 


6.7 


6.6 


6.5 


380 


5.8 


6.1 


6.3 


6.4 


6.6 


6.7 


6.6 


6.6 


6.7 


6.8 


6.7 


6.6 


6.5 


400 


5.9 


6.0 


6.2 


6.3 


6.4 


6.5 


6.6 


6.6 


6.5 


6.6 


6.6 


6.5 


6.4 


420 


6.1 


6.3 


6.2 


6.4 


6.3 


6.4 


6.5 


6.6 


6.5 


6.5 


6.5 


6.5 


6.4 


440 


6.3 


6.4 


6.4 


6.6 


6.5 


6.6 


6.5 


6.5 


6.5 


6.5 


63 


6.3 


6.2 


460 


6.7 


6.5 


6.5 


6.6 


6.7 


6.9 


6.7 


6.6 


6.6 


6.6 


6.5 


6.3 


6.2 


480 


7.1 


7.1 


7.0 


6.9 


6.9 


6.9 


7.0 


7.0 


6.8 


6.7 


6.6 


6.5 


6.3 


500 


7.4 


7.5 


7.4 


7.4 


7.3 


7.2 


7.3 


7.2 


7.1 


6.9 


6.8 


6.8 


6.6 


520 


7.9 


7.8 


7.8 


7.8 


7.8 


7.6 


7.6 


7.5 


7.5 


7.4 


7.1 


7.0 


6.9 


540 


8.3 


8.3 


8.3 


8.2 


8.2 


8.1 


8.0 


7.9 


7.9 


7.8 


7.6 


7.5 


7.2 


560 


8.2 


8.6 


8.4 


8.6 


8.7 


8.5 


8.5 


8.4 


8.2 


8.3 


8.2 


8.0 


7.6 


580 


8.2 


8.3 


8.6 


8.8 


8.8 


9.0 


8.9 


8.9 


8.7 


8.7 


8.6 


8.4 


8.4 


600 


7.7 


8.1 


8.5 


8.6 


8.9 


9.1 


9.1 


9.2 


9.2 


9.1 


9.0 


8.8 


8.7 


620 


7.4 


7.6 


8.0 


8.5 


8.7 


9.0 


9.2 


9.5 


9.5 


9.5 


9.4 


9.3 


9.2 


640 


6.7 


7.2 


7.5 


7.9 


8.3 


8.7 


9.0 


9.3 


9.5 


9.8 


9.8 


9.7 


9.7 


660 


6.0 


6.3 


7.0 


7.3 


7.7 


8.2 


8.7 


9.0 


9.4 


9.7 


9.8 


10.1 


10.0 


680 


4.9 


5.6 


6.0 


6.6 


7.1 


7.7 


8.1 


8.5 


9.0 


9.3 


9.8 


10.0 


10.2 


700 


4.2 


4.5 


5.2 


5.8 


6.4 


6.8 


7.4 


8.0 


8.5 


8.9 


9.2 


9.8 


10.1 


720 


3.6 


3.9 


4.3 


4.7 


5.3 


5.9 


6.6 


7.0 


7.8 


8.3 


8.8 


9.1 


9.7 


740 


3.1 


3.3 


3.6 


3.9 


4.4 


4.8 


5.6 


6.2 


6.9 


7.5 


8.0 


8.7 


9.2 


760 


2.8 


2.8 


3.0 


3.3 


3.6 


4.0 


4.4 


5.1 


5.8 


6.5 


7.2 


7.8 


8.4 


780 


2.5 


2.6 


2.5 


2.7 


3.1 


3.3 


3.7 


4.1 


4.8 


5.4 


6.1 


6.9 


7.6 


800 


2.7 


2.5 


2.5 


2.5 


2.5 


2.7 


3.0 


3.4 


3.8 


4.4 


5.0 


5.6 


6.6 


820 


2.9 


2.6 


2.4 


2.3 


2.2 


2.3 


2.6 


2.8 


3.1 


3.4 


4.1 


4.7 


5.4 


840 


3.1 


2.8 


2.6 


2.4 


2.3 


2.2 


2.3 


2.4 


2.6 


2.8 


3.2 


3.8 


4.3 


860 


3.6 


3.3 


3.0 


2.7 


2.4 


2.3 


2.1 


2.2 


2.3 


2.5 


2.7 


3.0 


3.4 


880 


4.3 


3.8 


3.6 


3.2 


2.8 


2.5 


2.3 


2.1 


2.0 


2.2 


2.3 


2.5 


2.6 


900 


4.9 


4.6 


4.2 


3.6 


3.4 


2.9 


2.6 


2.3 


2.2 


2.2 


2.1 


2.2 


2.4 


920 


5.4 


5.1 


4.6 


4.5 


3.9 


3.5 


3.2 


2.9 


2.6 


2.2 


2.0 


2.1 


2.2 


940 


5.9 


5.7 


5.3 


4.9 


4.7 


4.3 


3.8 


3.4 


3.0 


2.7 


2.4 


2.1 


2.0 


960 


6.5 


6.2 


5.9 


5.5 


5.1 


4.9 


4.5 


4.0 


3.4 


3.1 


2.8 


2.4 


2.3 i 


980 


6.9 


6.8 


6.4 


6.1 


5.8 


5.4 


5.1 


4.8 


4.3 


3.9 


3.5 


3.0 


2.7 


1000 


7.4 

200 


7.2 


7.0 


6.6 


6.4 


6.2 


5.7 


5.3 


4.9 


4.7 


4.1 


3.8 


3.4 


~ 


210 


220 


230 


240 


250 


260 


270 


280 


290 


300 


310 


320 1 



TAELE XXXI. 

Perturbations produced by Mars. 

Arguments II. and IV. 

IV. 



33 



|JL 


320 


] 330 


340 


350 


360 


I 370 


| 380 


: 390 


, 400 


| 410 


420 


430 


440 




,■ 


.. 


,, 


„ 


// 


l~ 


,, 


// 


/, 





3.4 


2.8 


2.6 


2.4 


2.2 


2.3 


2.3 


2.5 


2.7 


2.9 


3.4 


4.0 


4 5 


20 


3.9 


3.5 


3.1 


2.7 


2.6 


2.4 


2.4 


2.3 


2.5 


2.7 


3.0 


3.3 


3.S 


40 


4.7 


4.2 


3.9 


3.5 


3.0 


2.8 


2.7 


2.6 


2.5 


2.6 


2.8 


2.9 


3.2 


60 


5.4 


5.0 


4.6 


4.2 


3.8 


3.4 


3.1 


2.8 


2.8 


2.7 


2.7 


2.7 


3.0 


80 


6.0 


5.7 


5.4 


4.8 


4.4 


4.0 


3.6 


3.4 


3.1 


2.9 


2.9 


2.9 


2.9 


100 


6.6 


6.3 


5.9 


5.6 


5.2 


4.8 


4.3 


4.0 


3.7 


3.5 


3.2 


3.0 


3.0 


120 


7.0 


6.9 


6.4 


6.1 


5.8 


5.3 


5.2 


4.6 


4.3 


4.0 


3.8 


3.6 


3.4 


140 


7.4 


7.2 


6.9 


6.6 


6.5 


6.1 


5.6 


5.4 


5.0 


4.6 


4.3 


4.0 


3.9 


160 


7.6 


7.5 


7.3 


7.0 


6.8 


6.6 


6.2 


5.9 


5.5 


5.3 


4.9 


4.6 


4.4 


180 


7.8 


7.7 


7.5 


7.4 


7.3 


6.9 


6.7 


6.5 


6.2 


5.8 


5.6 


5.3 


5.0 


200 


7.9 


7.8 


7.7 


7.6 


7.5 


7.3 


7.1 


6.9 


6.6 


6.4 


6.1 


5.6 


5.5 


220 


7.7 


7.7 


7.7 


7.8 


7.7 


7.5 


7.3 


7.2 


7.0 


6.7 


6.5 


6.2 


5.9 


240 


7.6 


7.6 


7.6 


7.6 


7.7 


7.6 


7.5 


7.3 


7.2 


7.1 


6.9 


6.6 


6.4 


260 


7.4 


7.3 


7.5 


7.5 


7.5 


7.6 


7.6 


7.5 


7.5 


7.3 


7.1 


7.0 


6.7 


280 


7.3 


7.4 


7.3 


7.3 


7.4 


7.4 


7.3 


7.4 


7.3 


7.5 


7.2 


7.1 


6.9 


300 


7.1 


7.1 


7.1 


7.0 


7.2 


7.3 


7.3 


7.3 


7.2 


7.2 


7.3 


7.2 


7.1 


320 


6.8 


6.8 


6.9 


6.9 


6.8 


7.0 


7.1 


7.1 


7.1 


7.1 


.7.1 


7.0 


1 7.2 


340 


6,S 


6.7 


6.6 


6.6 


6.6 


6.8 


6.9 


6.9 


7.0 


7.0 6.9 


6.9 


1 6.9 


360 


6.5 


6.5 


6.4 


6.3 


6.4 


6.5 


6.6 


6.7 


6.8 


6.8 


6.8 


6.8 


6.9 


380 


6.5 


6.3 


6.3 


6.2 


6.2 


6.2 


6.3 


6.3 


6.4 


6.5 


6.6 


6.7 


6.7 


400 


6.4 


6.2 


6.2 


6.0 


6.1 


6.0 


6.0 


6.0 


6.0 


6.1 


6.2 


6.3 


6.4 


420 


6.4 


6.2 


6.1 


6?0 


5.9 


5.8 


5.9 


5.9 


5.9 


5.9 


5.9 


6.0 


6.0 


440 


6.2 


6.1 


6.0 


5.8 


5.8 


5.7 


5.6 


5.6 


5.6 


5.7 


5.7 


5.8 


5.9 


460 


6.2 


6.0 


5.9 


5.8 


5.7 


5.5 


5.5 


5.4 


5.5 


5.4 


5.5 


5.3 


5.4 


4S0 


6.3 


6.2 


6.0 


5.7 


5.6 


5.5 


5.4 


5.3 


5.2 


5.2 


5.2 


5.3 


5.3 


500 


6.6 


6.4 


6.2 


6.0 


5.7 


5.4 


5.3 


5.2 


5.1 


5.1 


5.1 


5.0 


5.0 


520 


6.9 


6.7 


6.4 


6.1 


6.1 


5.7 


5.5 


5.1 


5.1 


5.0 


4.9 


5.0 


4.9 


540 


7.2 


7.1 


6.7 


6.5 


6.2 


6.1 


5.8 


5.5 


5.2 


5.0 


4.9 


4.8 


4.S 


560 


7.6 


7.4 


7.3 


7.0 


6.6 


6.3 


6.0 


5.8 


5.4 


5.3 


5.0 


4.7 


4.7 


580 


84 


8.0 


7.8 


7.5 


7.0 


6.8 


6.3 


6.2 


5.9 


5.5 


5.3 


5.0 


4.9 


600 


8.7 


8.6 


8.3 


8.0 


7.8 


7.4 


7.0 


6.6 


6.3 


6.0 


5.6 


5.3 


5.1 


620 


9.2 


9.1 


8.9 


8.6 


8.4 


8.1 


7.6 


7.2 


6.8 


6.5 


6.1 


5.7 


5.3 


640 


9.7 


9.6 


9.4 


9.3 


9.0 


8.7 


8.2 


7.8 


7.4 


7.0 


6.6 


6.3 


5.8 


660 


10.0 


10.0 


9.9 


9.8 


9.6 


9.3 


8.9 


8.5 


8.2 


7.7 


7.2 


6.8 


6.4 


680 


10.2 


10.4 


10.3 


10.2 


10.1 


9.9 


9.6 


9.3 


9.0 


8.5 


8.1 


7.5 


7.1 


700 


10.1 


10.3 


10.5 


10.6 


10.4 


10.3 


10.1 


9.8 


9.6 


9.3 


8.9 


8.3 


7.8 


720 


9.7 


10.1 


10.3 


10.6 


10.7 


10.6 


10.5 


10.5 


10.2 


10.0 


9.6 


9.2 


8.6 


740 


9.2 


9.6 


10.0 


10.3 


10.6 


10.7 


10.S 


10.9 


10.6 


?0.5 


10.2 


9.9 


9.4 


760 


8.4 


9.0 


9.5 


9.8 


10.2 


10.6 


10.9 


11.0 


11.0 


11.0 


10.7 


10.5 


10.3 


780 


7.6 


8.2 


8.9 


9.4 


9.9 


10.3 


10.6 


10.9 


11.1 


11.2 


11.0 


10.S 


10.7 


800 


6.6 


7.3 


7.9 


8.5 


9.2 


9.8 


10.1 


10.6 


10.8 


11.1 


11.3 


ii.i ! 


11.0 


820 


5.4 


6.0 


7.0 


7.6 


8.2 


8.9 


9.6 


10.0 


10.5 


10.8 


11.0 


11.3 


11.3 


840 


4.3 


5.0 


5.6 


6.5 


7.2 


7.9 


9.8 


9.2 


9.9 


10.3 


10 7 


10.9 


11.2 


860 


3.4 


4.0 


4.6 


5.3 


6.1 


6.9 


7.5 


8.4 


9.1 


9.6 


10.1 


10.7 


10.9 


830 


2.6 


3.1 


3.7 


4.3 


5.0 


5.7 


6.6 


7.1 


8.1 


8.7 


9.4 


9.8 


10.4 


900 


2.4 


2.7 


3jO 


3.4 


4.0 


4.6 


5.4 


6.1 


6.9 


7.6 


8.4 


9.1 


9.7 


920 


2.2 


2.3 


2.3 


2.8 


3.3 


3.7 


4.3 


5.0 


5.8 


6.5 


7.2 


8.0 


8.7 


i 040 


2.0 


2.1 


2.3 


2.3 


2.7 


2.9 | 


3.4 


4.1 


4.7 


5.5 


6.1 


7.0 


7.7 


! 960 


2.3 


2.2 


2.2 


2.3 


2.3 


2.5 


2.8 


3.2 


3.9 


4.5 


5.1 


5.7 


6.5 


! 980 


2.7 


2.4 


2.2 


2.3 


2.3 


2.4 


2.5 


2.8 


3.0 


3.6 


4,1 


4.7 


5.5 


| 1000 


3.4 


2.8 


2.6 


2.4 
350 


2.2 
360 


2.3 ! 
370 


2.3 

380 


2.5 


2.7 
400 


2.9 
410 


3.4 
420 


4.0 


4.5 

440 

i 


320 


330 


340 


390 


430 



34 



ABLE XXXI. 



Perturbations produced by Mars. 

Arguments II and IV. 

IV. 



n. 




440 


450 


460 


'470 


480 


490 


500 


510 


520 


530 


540 


550 


560 


4.5 


5.2 


5.9 


6.6 


7.3 


8.0 


8.5 


9.0 


9.5 


10.0 


10.4 


10.7 


10.9 


20 


3.8 


4.3 


4.9 


5.6 


6.2 


6.9 


7.6 


8.2 


8.8 


9.3 


9.7 


10.0 


11.4 


40 


3.2 


3.7 


4.2 


4.S 


5.4 


5.9 


6.6 


7.3 


7.9 


8.4 


8.9 


9.4 


9.8 1 


60 


3.0 


3.2 


3.6 


4.0 


4.5 


5.1 


5.7 


6.3 


6.9 


7.5 


8.0 


8.6 


9.1 


80 


2.9 


3.1 


3.3 


3.5 


3.9 


4.4 


4.9 


5.4 


5.9 


6.5 


7.1 


7.7 


8.2 


! ioo 


3.0 


3.1 


3.2 


3.5 


3.6 


3.8 


4.2 


4.8 


5.3 


5.9 


6.4 


6.9 


7.4 


120 


3.4 


3.3 


3.3 


3.4 


3.5 


3.6 


3.9 


4.2 


4.7 


5.1 


5.6 


6.0 


6.6 


140 


3.9 


3.8 


3.6 


3.6 


3.6 


3.7 


4.0 


4.0 


4.2 


4.6 


5.0 


5.4 


5.9 


160 


4.4 


4.2 


3.9 


4.1 


3.8 


3.7 


4.0 


4.1 


4.2 


4.5 


4.6 


4.9 


5.3 


180 


5.0 


4.8 


4.4 


4.2 


4.2 


4.2 


4.0 


4.1 


4.3 


4.4 


4.4 


4.7 


5.0 


200 


5.5 


5.2 


5.1 


4.8 


4.6 


4.5 


4.5 


4.4 


4.5 


4.5 


4.7 


4.6 


4.8 


220 


5.9 


5.7 


5.5 


5.3 


5.1 


4.9 


4.9 


4.8 


4.7 


4.8 


4.8 


4.9 


5.0 


240 


6.4 


6.2 


5.9 


5.8 


5.6 


5.4 


5.3 


5.2 


5.1 


5.1 


5.1 


5.2 


5.2 


260 


6.7 


6.6 


6.4 


6.1 


6.0 


5.9 


5.8 


5.7 


5.6 


5.5 


5.4 


5.4 


5.4 


280 


6.9 


6.8 


6.7 


6.5 


6.3 


6.2 


6.1 


6.0 


5.9 


5.9 


5.9 


5.8 


5.8 


300 


7.1 


7.0 


6.8 


6.8 


6.6 


6.5 


6.4 


6.3 


6.2 


6.2 


6.2 


6.2 


6.2 


320 


7.2 


7.1 


6.9 


6.8 


6.8 


6.7 


6.6 


6.5 


6.5 


6.5 


6.5 


6.6 


6.6 


340 


6.9 


6.9 


7.0 


6.9 


6.9 


6.8 


6.7 


6.8 


6.7 


6.6 


6.7 


6.8 


6.9 


360 


6.9 


6.8 


6.8 


6.8 


6.8 


6.7 


6.7 


6.6 


6.6 


6.8 


6.8 


6.8 


6.9 


380 


6.7 


6.5 


6.5 


6.6 


6.7 


6.6 


6.6 


6.7 


6.7 


6.7 


6.8 


6.9 


6.9 


400 


6.4 


6.4 


6.3 


6.3 


6.4 


6.5 


6.5 


6.5 


6.6 


6.7 


6.7 


6.8 


6.8 


420 


6.0 


6.2 


6.3 


6.3 


6.2 


6.2 


6.3 


6.3 


6.3 


6.3 


6.5 


6.6 


6.7 


440 


5.9 


5.9 


6.0 


6.0 


6.0 


6.0 


6.0 


6.1 


6.0 


6.1 


6.2 


6.2 


6.4 


460 


5.4 


5.5 


5.7 


5.8 


5.S 


5.8 


5.8 


5.8 


5.8 


5.8 


5.9 


6.0 


6.1 


480 


5.3 


5.3 


5.5 


5.5 


5.5 


5.6 


5.5 


5.6 


5.4 


5.6 


5.7 


5.5 


5.8 


500 


5.0 


'5.0 


5.1 


5.2 


5.3 


5.3 


5.3 


5.2 


5.2 


5.2 


5.3 


5.4 


5.4 


520 


4.9 


4.9 


4.9 


4.8 


5.0 


5.1 


5.1 


5.1 


5.1 


5.1 


5.0 


5.0 


5.1 


540 


4.8 


4.8 


4.7 


4.8 


4.8 


4.9 


4.9 


5.0 


4.9 


4.8 


4.8 


4.9 


4.8 


560 


4.7 


4.6 


4.6 


4.7 


4.7 


4.6 


4.7 


4.7 


4.7 


4.7 


4.6 


4.6 


4.6 


580 


4.9 


4.6 


4.5 


4.5 


4.6 


4.5 


4.4 


4.4 


4.5 


4.5 


4.5 


4.4 


4.4| 


600 


5.1 


4.9 


4.6 


4.5 


4.4 


4.4 


4.4 


4.3 


4.3 


4.3 


4.3 


4.3 


4,3 


620 


5.3 


5.1 


4.9 


4.7 


4.6 


4.4 


4.3 


4.1 


4.2 


4.2 


4.2 


4.2 


4.1 


640 


5.8 


5.4 


5.2 


5.0 


4.7 


4.6 


4.4 


4.1 


4.1 


4.1 


4.2 


4.2 


4.0 


660 


6.4 


6.0 


5.7 


5.4 


.5.0 


4.8 


4.7 


4.5 


4.3 


4.2 


4.2 


4.1 


4.0 


680 


7.1 


6.6 


6.2 


5.7 


5.4 


5.1 


4.9 


4.7 


4.5 


4.4 


4.3 


4.0 


3.9 


700 


7.8 


7.2 


6.8 


6.4 


6.0 


5.6 


5.3 


5.0 


47 


4.6 


4.6 


4.3 


4.1 


720 


86 


8.0 


7.6 


7.1 


6.6 


6.2 


5.7 


5.5 


5.2 


4.9 


4.6 


4.6 


4.3 


740 


9.4 


9.0 


S.4 


8.0 


7.4 


6.9 


6.3 


6.0 


5.6 


5.3 


5.0 


4.*7 


4.5 


760 


10.3 


9.7 


9.3 


8.6 


8.1 


7.6 


7.2 


6.5 


6.2 


5.8 


5.5 


5.2 


4.9 


780 


10.7 


10.5 


9.9 


9.6 


9.0 


8.5 


7.8 


7.4 


7.0 


6.4 


6.1 


5.7 


5.5 


800 


11.0 


11.0 


10.6 


10.2 


9.9 


9.3 


8.8 


8.1 


7.7 


7.3 


6.7 


6.3 


5.8 


| 820 


11.3 


11.1 


10.9 


10.6 


10.3 


10.0 


9.6 


9.1 


8.5 


7.9 


7.4 


7.0 


6.6 


840 


11.2 


11.3 


11.2 


11.1 


11.0 


10.7 


10.2 


9.9 


9.4 


8.8 


8.2 


7.7 


7.3 


860 


10.9 


11.1 


11.4 


11.3 


11.3 


11.2 


10.7 


10.4 


9.9 


9.6 


9.2 


8.5 


7.9 


880 


10.4 


10.8 


11.0 


11.3 


11.2 


11.2 


11.2 


10.9 


10.5 


10.3 


9.8 


9.3 


8.7 


900 


9.7 


10.1 


10.6 


11.0 


11.2 


11.2 


11.2 


11.0 


10.9 


10.7 


10.2 


10.0 


9.4 


920 


8.7 


9.3 


9.9 


10.3 


10.8 


11.0 


11.1 


11.2 


11.2 


11.0 


10.7 


10.4 


10.1 


940 


7.7 


8.2 


8.8 


9.5 


10.1 


10.4 


10.9 


11.0 


11.2 


11.2 


11.0 


10.7 


10.5 


960 


6.5 


7.3 


8.1 


8.6 


9.3 


9.8 


10.2 


10.6 


10.8 


11.1 


11.2 


10.9 


10.8 


980 


5.5 


6.2 


7.0 


7.7 


8.3 


8.9 


9.5 


10.0 


10.4 


10.6 


10.8 


11.0 


10.9 


1000 


4.5 


5.2 


5.9 


6.6 


7.3 


8.0 


8.5 


9.0 


9.5 


10.0 


10.4 


10.7 


10.9 




440 


450 


460 


470 


480 


490 


500 


510 


520 


530 


540 


550 


560 



TABLE XXXI. 



35 



Perturbations produced by Mars, 

Arguments II and IV. 

IV. 



1- 


560 j 570 


5S0 


590 


600 


610 


620 


630 


640 


650 


1 660 


670 


680 


i 


"T~iT7~ 


// 




" 




,- 


// 


// 




\~ 


„ 


,, 





10 9' 10.8 


10.6 


10.4 


10.3 


10.0 


9.7 


9.2 


8.9 


8.5 


8.1 


7.9 


7.7 


20 


11.4 10.6 


10.7 


10.6 


10.4 


10.2 


9.9 


9.7 


9.3 


9.0 


8.8 


8.5 


'8.1 


40 


•9.8110.1 


10.4 


10.4 


10.5 


10.3 


10.2 


9.9 


9.6 


9.4 


9.1 


8.9 


8.5 


60 


9.1 


1 9.4 


9.8 


10.2 


10.2 


10.3 


10.2 


10.1 


9.9 


9.6 


93 


9.0 


8.8 


80 


8.2 


8.7 


9.0 


9.3 


9.6 


9.8 


10.0 


9.9 


9.8 


9.7 


9.5 


9.3 


91 


100 


7.4 


7.9 


8.4 


8.7 


9.0 


9.4 


9.6 


9.7 


9.8 


9.7 


9.7 


9.5 


9.2 


120 


6.6 


6.9 


7.6 


8.1 


8.3 


8.6 


9.0 


9.2 


9.4 


9.5 


9.5 


9.4 


9.3 


140 


5.9 


6.3 


6.8 


7.2 


7.7 


8.0 


8.3 


8.7 


8.9 


9.1 


9.2 


9.3 


9.3 


160 


5.. 3 


5.8 


6.0 


6.5 


6.9 


7.4 


1 7.7 


8.0 


8.4 


8.5 


8.8 


8.9 


9.0 


ISO 


5.0 


5.2 


5.6 


6.0 


6.3 


6.7 


7.1 


7.2 


7.7 


8.1 


8.1 


8.4 


8.6 


200 


4.8 


5.0 


5.3 


5.4 


5.8 


6.1 


6.5 


6.7 


7.1 


7.3 


7.7 


7.8 


8.0 


220 


5.0 


5.0 


5.1 


5:3 


5.5 


5.7 


6.0 


6.3 


6.6 


6.8 


7.0 


7.3 


7.5 


240 


5.2 


5.2 


5.3 


5.3 


5.4 


5.5 


5.7 


5.9 


6.1 


6.4 


6.6 


6.8 


7.1 


260 


5.4 


5.5 


5.5 


5.5 


5.5 


-5.5 


5.5 


5.7 


5.8 


6.0 


6.3 


6.4 


6.5 


280 


5.8 


5.8 


5.8 


5.9 


5.8 


5.8 


5.8 


5.9 


5.9 


5.9 


6.0 


6.1 


6.2 


300 


6.2 


6.1 


6.2 


6.1 


6.1 


6.1 


6.2 


6.1 


6.0 


5.9 


5.9 


6.0 


6.1 ! 


320 


6.6 


0.5 


6.6 


6.6 


6.5 


6.5 


6.6 


6.5 


6.5 


6.3 


6.1 


6.0 


6.0 


340 


6.9 


6.9 


6.9 


7.0 


7.0 


6.9 


6.8 


6.9 


6.9 


6.8 


6.6 


€.5 


6.3 


360 


6.9 


7.0 


7.2 


7.3 


7.3 


7.3 


7.4 


7.3 


7.3 


7.1 


7.1 


7.0 


6.7 


380 


6.9 


7.0 


7.2 


7.4 


7.5 


7.6 


7.7 


7.7 


7.7 


7.6 


7.5 


7.4 


7.2 


400 


6.8 


7.0 


7.1 


7.3 


7.6 


7.9 


8.0 


8;0 


8.1 


8.1 


8.1 


7.9 


7.8 


420 


6.7 


•Q.9 


7.0 


7.2 


7.6 


7.8 


8.0 


8.2 


8.3 


8.4 


8.4 


8.5 


8.4 


440 


6.4 


6.6 


6.9 


7.0 


7.3 


7.5 


7.9 


8.2 


8.4 


8.6 


8.8 


8.8 


8.9 


460 


6.1 


6.2 


6.5 


6.9 


7.1 


7.2 


7.6 


8.0 


8.4 


8.7 


9.0 


9.1 


9.2 


4S0 


5.8 


5.9 


6.0 


6.2 


6.7 


7.1 


7.2 


7.6 


7.9 


8.5 


8.9 


9.2 


9.3 


500 


5.4 


5.5 


5.6 


5.9 


6.1 


6.4 


6.9 


7.2 


7.7 


7.9 


8.4 


9.0 


9.4 


520 


5.1 


5.2 


5.2 


5.3 


5.6 


5.9 


6.3 


6.7 


7.0 


7.6 


8.0 


8.4 


9.0 


540 


4.8 


4.8 


4.S 


5.0 


5.1 


5.4 


5.6 


6.0 


6.4 


6.7 


7.5 


8.1 


8.5 


560 


4.6 


4.5 


4.5 


4.5 


4.7 


4,8 


5.0 


5.3 


5.8 


6.2 


6.6 


7.1 


7.8 


580 


4.4 


4.3 


4.3 


4.3 


4.3 


4.3 


4.5 


4.7 


5.2 


5.5 


5.9 


6.4 


6.9 


600 


4.3 


4.3 


4.2 


4.1 


4.0 


4.0 


4.1 


4.2 


4,5 


4.8 


5.1 


5.7 


6.2 


620 


4.1 


4.0 


4.0 


3.9 


3.9 


3.8 


3.8 


3.8 


3.8 


4.0 


4.4 


4.9 


5.4 


640 


4.0 


3.9 


4.0 


3.8 


3.8 


3.8 


3.7 


3.5 


3.5 


3.6 


3.8 


4.0 


4.5 


660 


4.0 


4.0 


3.9 


3.8 


3.7 


3.5 


3.5 


3.4 


3.3 


3.3 


3.4 


3.5 


3.7 


680 


3.9 


4.0 


3.9 


3.8 


3.6 


3.5 


3.4 


3.3 


3.2 


3.1 


3.1 


3.1 


3.1 


700 


4.1 


3.9 


3.9 


3.9 


3.7 


3.5 


3.4 


3.3 


3.2 


3.0 


3.0 


3.0 


2.9 


720 


4.3 


4.1 


4.0 


3.9 


3.8 


3.8 


3.5 


3.4 


3.1 


2.9 


2.9 


2.7 


2.7 


740 


4.5 


4.2 


4.2 


4.2 


4.0 


3.7 


3.6 


3.4 


3.3 


3.0 


2.8 


2.6 


2.5 


760 


4.9 


4.7 


4.5 


4.3 


4.2 


4.1 


3.8 


3.0 


3.3 


3.1 


2.9 


2.8 


2.5 


780 


5.5 


5.1. 


4.9 


4.5 


4.4 


4.3 


4.1 


3.9 


3.8 


3.4 


3.2 


3.0 


2.7 


800 


5.8 


5.6 


5.2 


5.0 


4.6 


4.5 


4.4 


4.3 


4,1 


3.8 


3.5 


3.1 


2.8 


820 


6.6 


6.1 


5.8 


5.5 


5.3 


5.0 


4.8 


4.6 : 4.4 


4.2 


4.0 


3.6 


3.3 


840 


7.3 


6.8 


6.5 


6.1 


5.7 


5.5 


5.2 


5.0 


4,7 


4.6 


4.3 


4.1 


3.8 


860 


7.9 


7.5 


7.0 


6.7 


6.4 


5.9 


5.8 ■■ 


5.4 


5.1 


5.0 


4.8 


4.6. 


4.4 


880 


8.7 


8.2 


7.8 


7.3 


6.9 


6.6 


6.3 


6.0 


5.7 


5.4 


5.2 


5.0 


4.7 


900 


9.4 


9.0 


8.5 


8.0 


7.6 


7.2 


6.8. 


6.6 


6.3 


5.9 


5.6 


5.4 


5.2 


920 


10.1 


9.8 


9.2 


8.7 


8.3 


7.8 


7.4 


7.0 


6.7 


6.4 


6.0 


5.8 


5.7 


940 


10.5 


10.2 


9.8 


9.4 


8.8 


8.5 


8.0 


7.6 


7.3 


6.9 1 


6.6 


6.2 


G 1 


960 


10.8 


10.5 


10.2 


10.0 


9.5 


9.1 


8.6 


8.2 


7.8 


7.5! 


7.1 


6.8 


66 


980 


10.9 


10.7 


10.3 


10.2 


9.9 


9.6 


9.2 


9.0 


8.5 


8.0 


7.7 


7.4 


7.2 


K)00 


10.9 


10.8 


10.6 


10.4 


10.3 


10.0 


9.7 


9.2 


8.9 


8.5 


8.1 


7.9 


7.7 






560 


570 


580 


590 


600 


610 


620 


630 


640 1 


650 ! 


660 


670 


680 



36 



TABLE XXXI. 



Perturbations produced by Mars. 

Arguments II. and IV. 

IT. 



El. 


680 

7.7 


690 

7.4 


700 

6.9 


710 

6.8 


720 


730 


740 


750 


760 


770 


780 


790 


800 


o 


6.7 


6.4 


6.1 


5.8 


5.5 


5.2 


4.8 


4.4 


fr 

3.7 


20 


8.1 


7.8 


7.4 


7.0 


7.1 


6.9 


6.7 


6.4 


6.1 


5.8 


5.5 


5.1 


4.7 


40 


8.5 


8.3 


7.8 


7.5 


7.2 


7.1 


7.0 


6.9 


6.6 


6.4 


6.1 


5.8 


5.3 


60 


8.8 


8.6 


8.3 


8.1 


7.8 


7.6 


7.5 


7.4 


7.1 


6.9 


6.7 


6.3 


6.0 


80 


9.1 


8.9 


8.7 


8.4 


8.1 


8.0 


7.8 


7.6 


7.4 


7.3 


7.1 


6.9 


6.5 


100 


9.2 


8.9 


8.8 


8.7 


8.6 


8.3 


8.0 


7.7 


7.6 


7.6 


7.6 


7.3 


7.0 


120 


9.3 


9.2 


9.0 


8.7 


S.6 


8.4 


8.2 


8.1 


7.9 


7.8 


7.7 


7.6 


7.5 


140 


9.3 


9.2 


9.0 


9.0 


8.7 


8.5 


8.4 


8.3 


8.0 


7.8 


7.7 


7.7 


7.7 


160 


9.0 


9.0 


8.9 


8.8 


8.7 


8.6 


8.5 


8.4 


8.2 


8.0 


7.& 


' 7.8 


78 


180 


8.6 


8.6 


8.7 


8.7 


8.7 


8.6 


8.5 


S.3 


8.3 


8.0 


8.2 


7.8 


7.9 


200 


8.0 


8.2 


8.3 


8.3 


8.5 


8.4 


8.4 


8.4 


8.2 


8.1 


8.1 


8.1 


7.9 


220 


7.5 


7.7 


7.9 


8.1 


8.2 


8.2 


8.1 


8.2 


8.2 


8.0 


8.1 


8.0 


8.0 


240 


7.1 


7.2 


7.4 


7.5 


7.6 


7.7 


7.8 


7.8 


7.9 


8.0 


8.0 


7.8 


7.8 


260 


6.5 


6.7 


6.9 


7.1 


7.2 


7.3 


7.4 


7.5 


7.6 


7.6 


7.7 


7.7 


7.8 


280 


6.2 


6.3 


6.5 


6.7 


6.7 


6.9 


7.1 


7.2 


7.3 


7.3 


7.3 


7.3 


7.4 


300 


6.1 


6.0 


6.2 


6.4 


6.4 


6.5 


6.6 


6.7 


6.9 


6.9 


6.9 


7.1 


7.1 


320 


6.0 


6.0 


6.0 


6.0 


6.2 


6.1 


6.2 


6.3 


6.5 


6.5 


6.6 


6.6 


6.8 


340 


6.3 


6.2 


6.0 


6.0 


6.0 


6.0 


6.1 


6.1 


6.2 


6.2 


6.3 


6.3 


6.4 


360 


6.7 


6.6 


6.4 


6.1 


6.0 


5.9 


6.0 


5.9 


5.9 


5.9 


6.0 


6.1 


6.2 


380 


7.2 


7.1 


6.8 


6.& 


6.4 


6.2 


6.1 


5.9 


5.8 


5.7 


5.6 


5.8 


5.9 


400 


7.8 


7.7 


7.4 


7.1 


6.8 


6.6 


6.4 


6.1 


6.0 


5.8 


5.6 


5.5 


5.6 


420 


8.4 


8.2 


8.0 


7.8 


7.5 


7.2 


6.8 


6.5 


6.2 


60 


5.7 


5.5 


5.4 


440 


8.9 


8.8 


8.7 


8.4 


8.2 


7.8 


7.5 


7.1 


6.6 


6.2 


6.0 


5.7 


5.6 


460 


9.2 


9.2 


9.2 


9.0 


8.8 


8.5 


8.2 


7.9 


7.5 


6.9 


6.5 


6.3 


6.0 


480 


9.3 


9.5 


9.6 


9.6 


9.4 


9.2 


9.1 


8.6 


8.3 


7.8 


7.2 


6.9 


6.5 


500 


9.4 


9.6 


9.8 


10.0 


9.9 


9.8 


9.6 


9.4 


9.1 


8.7 


8.2 


7.6 


7.2 


520 


9.0 


9.5 


9.8 


10.1 


10.2 


10.3 


10. 3 


10.0 


9.8 


9.5 


9.1 


8.5 


8.0 


540 


8.5 


9.1 


9.5 


10.0 


10.3 


10.5 


10.6 


10.6 


10.4 


10.1 


9.8 


9.5 


9.0 


560 


7.8 


8.5 


9.0 


9.5 | 


9.9 


10.4 


10.8 


10.8 


10.9 


10.8 


10.6 


10.2 


9.9 


5S0 


6.9 


7.6 


8.3 


9.0 


9.7 


10.0 


10.4 


10.7 


11.1 


11.2 


11.0 


11.0 


10.6 


600 


6.2 


6.8 


7.4 


8.0 


8.9 


9.6 


10.1 


10.4 


10.9 


11.3 


11.4 


11.3 


11.2 


620 


5.4 


5.9 


6.5 


7.1 


7.8 


8.6 


9.4 


10.3 


10.6 


11.0 


11.5 


11.7 


11.7 


640 


4.5 


5.0 


5.5 


6.2 


6.8 


7.6 


8.4 


9.2 


10.0 


10.7 


11.1 


11.6 


11.8 


660 


3.7 


4.1 


4.7 


5.2 


5.9 


6.5 


7.3 


8.3 


9.1 


9.8 


10.5 


11.2 


11.5 


680 


3.1 


3.4 


3.8 


4.3 


4.8 


5.5 


6.2 


7.0 


7.8 


8.7 


9.6 


10.2 


11.0 


700 


2.9 


2.8 


3.0 


3.4 


3.9 


4.5 


5.2 


6.0 


6.7 


7.5 


8.5 


9.4 


10.1 


720 


2.7 


2.6 


2.5 


2.7 


3.1 


3.5 


4.0 


4.8 


5.6 


6.4 


7.3 


8.2 


9.1 


740 


2.5 


2.4 


2.4 


2.4 


25 


2.7 


3.1 


3.6 


4.5 


5.2 


6.1 


6.9 


7.8 


760 


2.5 


2.3 


2.2 


2.1 


2.1 


2.3 


2.4 


.2.8 


3.2 


4.1 


4.7 


5.7 


6.6 


780 


2.7 


2.5 


2,3 


2.1 


2.0 


1.9 


2.1 


2.2 


2.5 


2.9, 


3.6 


4.4 


5.2 


800 


2.8 


2.7 


2.4 


2.2 


2.0 


1.8 


1.8 


1.8 


2.0 


2.3 


2.5 


3.2 


4.0 


820 


3.3 


3.0 


2.7 


2.3 


2.1 


1.9 


1.8 


1.5 


1.7 


1.7 


2.0 


2.2 


2. 9 


840 


3.8 


3.5 


3.0 


2.6 


2.3 


2.1 


1.9 


1.6 


1.5 


1.5 


1.6 


1.7 


2.2 


860 


4.4 


4.0 


3.5 


3.2 


2.8 


2.3 


1.9 


1.7 


1.4 


1.3 


1.2 


1.4 


1.6 


880 


47 


4.4 


4.1 


3.7 


3.3 


3.0 


2.5 


2.1 


1.7 


1.4 


1.3 


1.2 


1.2 


900 


5.2 


5.0 


4.6 


4.3 


4.0 


3.6 


3.2 


2.7 


2.2 


1.6 


1.3 


1.2 


1.1 


! 920 


5.7 


5.3 


51 


5.0 


4.6 


4.2 


3.8 


3.4 


2.9 


2.3 


1.9 


1.3 


1.1 


940 


6.1 


5.9 


5.6 


5.4 


5.2 


4.8 


4.5 


3.9 


3.5 


3.1 


2.6 


2.1 


1.5 


960 


6.6 


6.4 


6.2 


5.9 


5.6 


5.4 


5.1 


4.7 


4.3 


3.7 


3.2 


2.8 


2.3 


980 


7.2 


6.9 


6.6 


6.4 


6.2 


5.9" 


5.6 


5.3 


5.0 


4.6 


4.0 


3.5 


3.0 


1000 


7.7 


7.4 


6.9 


e.s 


6.7 


6.4 


6.1 


5.8 


5.5 


5.2 


4.8 


4.4 


3.7 




680 


690 


\ 700 


710 


720 


730 


740 


750 


760 


770 


780 


790 1 


800 


I 

























TABLE XXXI. 



37 



Perturbations produced hy Mars. 

Arguments II. and IV. 

IV. 



II. 


$00 


| 810 


820 


830 


1 840 

1 " 


850 


860 


870 


880 


890 


900 


, 910 

I " 


920 




,, 


// 


// 


,/ 


// 


„ 


/, 





3.7 


3.2 


2.6 


2.1 


1.7 


1.3 


0.9 


0.7 


0.7 


1.0 


1.2 


1 1.6 


2.2 


20 


4.7 


4.2 


3.6 


3.1 


2.4 


1.9 


1.5 


1.2 


0.8 


0.6 


0.9 


1.2 


15 


40 


5.3 


4.9 


4.5 


3.8 


3.3 


2.7 


2.0 


1.7 


1.4 


1.0 


0.8 


! 0.9 


1.0 


60 


6.0 


5.7 


5.2 


4.7 


4.1 


3.6 


3.1 


2.6 


2.0 


1.5 


1.2 


0.9 


1.0 


80 


6,5 


6.3 


6.0 


5.5 


5.0 


4.6 


4,0 


3.4 


2.7 


2.2 


1.8 


| 1.5 


1.3 


100 


7.0 


6.7 


6.5 


6.3 


5.9 


5.3 


4.9 


4.4 


3.7 


3.1 


2.5 


I 2.1 


1.7 


120 


7.5 


7.3 


7.0 


6.8 


6.5 


6.2 


5.7 


5.1 


4.7 


4.1 


3.5 


2.9 


2.4 


140 


7.7 


7.7 


7.5 


7.3 


7.0 


6.7 


6.4 


6.0 


5.6 


5.1 


4.5 


3.8 


33 


160 


7.8 


7.9 


7.7 


7.6 


7.4 


7.2 


7.0 


6.8 


6.3 


5.8 


5.4 


4.8 


4.2 


180 


7.9 


7.8 


7.9 


7.9 


7.7 


7.6 


7.5 


7.1 


7.0 


6.6 


6.1 


5.7 


5.2 


200 


7.9 


7.9 


7.8 


7.9 


7.8 


7.7 


7.6 


7.5 


7.5 


7.1 


6.8 


6.3 


6.1 


220 


8.0 


7.9 


7.8 


7.8 


7.8 


7.8 


7.8 


7.8 


7.6 


7.5 


7.4 


7.1 


6.7 


240 


7.8 


7.7 


7.7 


7.7 


7.7 


7.7 


7.8 


7.8 


7.7 


7.6 


7.6 


7.5 


7.2 


260 


7.8 


7.7 


7.7 


7.6 


7.7 


7.7 


7.7 


7.7 


7.7 


7.7 


7.8 


7.8 


7.6 


280 


7.4 


7.4 


7.5 


7.5 


7.5 


7.5 


7.5 


7.5 


7.5 


7.6 


7.6 


7.8 


7.7 


300 


7.1 


7.2 


7.3 


7.3 


7.3 


7.3 


7.3 


7.4 


7.5 


7.4 


7.5 


7.5 


7.7 


320 


6.8 


6.9 


6.8 


7.0 


7.1 


7.1 


7.1 


7.1 


7.3 


7.3 


7.3 


7.4 


7.4 


340 


6.4 


6.5 


6.6 


6.6 


6.7 


6.7 


6.8 


6.9 


7.0 


7.1 


7.2 


7.2 


7.2 


360 


6.2 


6.2 


6.2 


6.3 


6.4 


6.4 


6.5 


6.6 


6.7 


6.7 


6.9 


6.9 


7.1 


380 


5 9 


5.8 


5.8 


5.9 


6.0 


6.1 


6.2 


6.3 


6.4 


6.4 


6.4 


6.6 


6.8 


400 


5.6 


5.6 


5.6 


5.7 


5.7 


5.7 


5.8 


5.9 


5.9 


6.0 


6.1 


6.2 


6.4 


420 


5.4 


5.4 


5.5 


5.5 


5.5 


5.5 


5.5 


5.5 


5.6 


5.6 


5.6 


5.7 


5.8 


440 


56 


5.3 


5.3 


5.3 


5.3 


5.2 


5.2 


5.2 


5.2 


5.1 


5.0 


5.3 


5.5 


460 


6.0 


5.6 


5.4 


5.3 


5.2 


5.2 


5.1 


5.0 


5.1 


5.2 


5.2 


5.2 


5.3 


4S0 


6.5 


6.0 


5.7 


5.4 


5.2 


5.2 


5.1 


4.9 


4.9 


4.9 


4.9 


5.0 


5.0 


500 


7.2 


6.8 


6.3 


5.9 


5.6 


5.3 


5.0 


4.8 


4.9 


4.8 


4.8 


4.8 


4.9 


520 


8.0 


7.4 


7.0 


6.5 


6.1 


5.5 


5.4 


5.1 


4.9 


4.7 


4.7 


4.7 


4.8 


540 


9.0 


8.4 


7.8 


7.3 


6.7 


6.3 


5.8 


5.4 


5.2 


4.9 


4.7 


4.7 


4.7 


560 


9.9 


9.5 


8.8 


8.2 


7.7 


7.1 


6.5 


6.0 


5.7 


5.3 


5.0 


4.8 


4.6 


580 


10.6 


10.2 


9.8 


9.3 


8.S 


8.1 


7.2 


6.8 


6.4 


6.0 


5.6 


5.1 


4,9 


600 


11.2 


11.0 


10.7 


10.3 


9.6 


9.1 


8.5 


7.7 


7.1 


6.4 


6.1 


5.6 


5.3 


620 


11.7 


11.5 


11,4 


11.0 


10.6 


9.9 


9.5 


8.9 


8.1 


7:4 


6.8 


6.3 


5.9 


640 


11.8 


11.9 


11.8 


11.7 


11.3 


11.0 


10.4 


9.8 


9.3 


8.5 


7.8 


7.1 


6.6 


660 


11.5 


11.8 


12.0 


12.1 


11.9 


11.6 


11.2 


10.8 


10.2 


9.6 


8.9 


8.2 


7.5 


680 


11.0 


11.6 


12.1 


12.2 


12.1 


12.2 


12.1 


11.5 


11.1 


10.6 


10.1 


9.2 


8.5 


700 


10.1 


10.9 


11.6 


12.1 


12.4 


12.3 


12.3 


12.3 


11.9 


11.4 


10.8 


10.4 


9.7 


720 


9.1 


10.0 


10.6 


11.4 


11.9 


12.4 


12.6 


12.5 


12.4 


12.0 


11.6 


11.2 


10.8 


740 


7.8 


8.8 


9.7 


10.5 


11.3 


11.8 


12.3 


12.8 


12.6 


12.6 


12.3 


11.9 


11.5 


760 


6.6 


7.6 


8.5 


9.4 


10.3 


11.0 


11.7 


12.1 


12.6 


12.8 


12.7 


12.5 


12.1 


780 


5.2 


6.3 


7.1 


8.1 


9.2 


10.1 


10.7 


11.6 


12.0 


12.4 


12.8 


12.9 


12.8 


800 


4.0 


4.8 


5.7 


6.7 


7.7 


8.7 


9.7 


10.5 


11.3 


11.9 


12.3 


12.5 


12.9 


820 


2.9 


3.6 


4.4 


5.4 


6.4 


7.3 


8.4 


9.5 


10.3 


11.0 


11.7 


12.1 


12.5 


840 


2.2 


2.7 


3.3 


4.0 


4.9 


6.0 


7.0 


8.0 


9.1 


10.0 


10.8 


11.4 


12.0 


860 


1.6 


1.6 


2.2 


2.9 


3.6 


4.6 


5.6 


6.6 


7.6 


8.6 


9.6 


10.5 


11.2 


880 


1.2 


1.3 


1.5 


1.9 


2.6 


3.3 


4.1 


5.2 


6.1 


7.1 


8.2 


9.2 


10.1 


900 


1.1 


1.1 


1.2 


1.3 


1.7 


2.2 


2.9 


3.8 


4.8 


5.7 


6.8 


7.9 


8.8 


920 


1.1 


1.0 


1.0 


1 i 


1.1 


1.4 


1.9 


2.G 


3.4 


4.4 


5.3 


6.3 


'7.4 


940 


1.5 


1.1 


0.8 


0.9 


1.0 


1.1 


1.3 


1.6 


2.3 


3.1 


3.9 


5.0 


5.9 


960 


23 


1.7 


1.3 


0.9 


0.7 


0.8 


0.9 


1.2 


1.4 


2.0 


2.8 


3.5 


4.6 


980 


3 


2.5 


1.9 


1.4 


1.2 


1.0 


0.8 


0.9 


1.2 


1.4 


1.7 


2.4 


3.3 


1000 


37 


3.2 

810 


2.6 


2.1 
830 


1.7 
840 


1.3 


0.9 


0.7 


0.7 


1.0 


1.2 


1.6 


2.2 


i 


800 


820 


850 


860 


870 


880 


890 


900 


910 


920 



3S 



TABLE XXXL 



TABLE XXXIL 



Perturbations by Mars. 
Arguments II. and IV. 
IV. 



Perfs. by Jupiter 
ArgV. II. and V. 
V. 



n. 


920 


930 


940 


950 


960 


970 


980 


990 


1000 


JL 


10 


20 


30 




,, 




,, 


~ 


,, 


// 


„ 


" 


~ 


" 


' ' 


T, 


„ 





2.2 


3.0 


3.S 


4.8 


5.S 


6.9 


7.8 


8.4 


9.5 


15.3 


15.1 


15.0 


15.0 


20 


1.5 


2.1 


2.6 


3.4 


4.4 


5.5 


6.5 


7.6 


8.7 


14.9 


14,9 


14.7 


14.8 


40 


1.0 


1.4 


l.S 


2.5 


3.2 


4.0 


5.2 


6.0 


7.1 


14.7 


14.6 14.6 


14.5! 


60 


1.0 


1.1 


1.3 


l.S 


2.3 


3.0 


3.7 


4.8 


5.8 


14.4 


14.4 14.4 


14.4 ! 


SO 


1.3 


1.1 


1.2 


1.4 


1.6 


2.2 


2.7 


3.6 


4.5 


13.4 


13.9 


14.0 


14.2J 


« 100 


1.7 


1.3 


1.2 


1.2 


1.3 


1.6 


2.0 


2.6 


3.3 


13.2 


13.4 


13.6 


13.7 


120 


2.4 


2.0 


1.5 


1.4 


1.4 


1.4 


1.7 


1.9 


2.4 


12.3 


12.7 


13.0 


13.3 


j 140 


3.3 


2.8 


2.3 


2.0 


1.7 


1.5 


1.5 


l.S 


2.1 


11.3 


11.8 


12.1 


12.5 


160 


4.2 


3.6 


3.1 


2.6 


2.1 


2.0 


1.7 


1.7 


1.9 


10.2 


10.7 


11.2 


11.7 


180 


5.2 


4.6 


4.0 


3.5 


3.1 


2.5 


2.0 


2.0 


1.9 


9.1 


9.6 


10.1 


10.7 


200 


0.1 


5.5 


5.0 


4.4 


3.9 


3.5 


2.S 


2.7 


2.9 


7.8 


S.3 


8.9 


9.5 


220 


6.7 


6 3 


5.8 


5.4 


4.9 


4 4 


3.9 


3.2 


3.0 


6.8 


7.2 


7.7 


8.3 


240 


7.2 


6.9 


5.6 


6 1 


5.6 


5 3 


4.8 


4.2 


3.7 


5.7 


6.2 


6.6 


7.2 


260 


7.6 


7.5 


7.1 


6.8 


6.5 


6.0 


5.6 


5.2 


4.S 


4.8 


5.2 


5.6 


6.1 


280 


7.7 




7.5 










5.9 


5.5 


3.9 


4.1 


4.7 


5.2 


300 


7.7 


7.7 


■ 7.7 


7.7 


7.4 


7.2 


7.0 


6.6 


6.1 


3.4 


3.5 


3.9 


4.3 


320 


7.4 


7.4 


7.6 


7.7 


7.6 


76 


7.3 


7.1 


6.9 


3.2 


3.1 


3.4 


3.6 


340 


7.2 


7.2 


7.3 


7.5 


7.7 


7.6 


7.6 


7.6 


7.7 


3.2 


3.0 


3.0 


3.1 


360 


7.1 


7.1 


7.1 


7.2 


7.2 


7.6 


7.6 


7.6 


7.5 


3.5 


3.2 


2.9 


2.9 


3S0 


6.8 


6.9 


7.0 


7.0 


7.0 


7.1 


7.3 


7.5 


7.5 


4.5 


4.0 


3.4 


3.1 


400 


6.4 


6.6 


6.6 


6.7 


6.7 


6.9 


7.0 


7.1 


7.3 


5.0 


4.3 


3.8 


3.5 


420 


5.8 


5.9 


6.2 


63 


6.6 


6.5 


6.7 


6.7 


6.9 


6.1 


5.2 


4.6 


4.1 


440 


5.5 


5.6 


5.7 


5.8 


6.0 


6.1 


6.3 


6.5 


6.5 


7.5 


6.6 


5.8 


4.9 


460 


5.3 


5.3 


5.4 


5.7 


5.7 


5.7 


5.9 


6.1 


6.2 


9.0 


7.9 


7.0 


6.3 


480 


5.0 


5.0 


5.0 


5.1 


5.3 


5.4 


5.5 


5.6 


5.8 


10.5 


9.5 


S.5 


7.6 


500 


4.0 


4.9 


5.0 


5.0 


5.0 


5.1 


5.2 


5.3 


5.3 


12.3 


11.3 


10.0 


9.1 


520 


4.8 


4.8 


4.8 


4.8 


4.8 


4.7 


4.9 


5.0 


5.1 


14.0 


12.7 


11.7 


10.7 


540 


4.7 


4.7 


4.6 


4.6 


4.6 


4.5 


4.6 


4.6 


4.7 


15.6 


145 


13.3 


12.3 


560 


4.6 


4.5 


4.5 


4.4 


4.5 


4.5 


4.5 


4.5 


4.4 


17.1 


16.1 


15.1 


14.0 


5S0 


4.9 


4.7 


4.6 


4.5 


4.4 


4.4 


4.4 


4.4 


4.2 


18.6 


17.4 


16.5 


15.7 


600 


5.3 


4.9 


4.8 


4.7 


4.5 


4,4 


4.4 


4.3 


4.1 1 


19.8 


19.0 


17.9 


17.0 


620 


5.9 


5.5 


5.1 


4.8 


4.6 


4.5 


4.4 


43 


4.2 


20.8 


20.1 


19.2 


18.4 


640 


6.6 


6.1 


5.6 


5.4 


5.0 


4.7 


4.6 


4.5 


4.3 


21.6 


20.9 


20.2 


19.5 


660 


7.5 


6.8 


6.3 


5.9 


5.5 


5.3 


4.9 


4.8 


4.6 


22.1 


21.6 


21.0 


20.4 


680 


8.5 


7.8 


7.3 


6.5 


6.1 


5.6 


5.4 


5.1 


4.8 


22.3 


22.0 


21.6 


21.2 


700 


9.7 


8.9 


8.1 


7.6 


7.0 


6.3 


5.9 


5.6 


5.3 


22.2 


22.0 


21.7 


21.5 


720 


10.8 


10.0 


9.3 


8.5 


7.9 


72 


6.6 


6.1 


5.8 


22.0 


21.9 


21.7 


21.6 


740 


11.5 


11.0 


10.2 


9.7 


8.9 


8.2 


7.6 


6.9 


6.5 


21.6 


21.6 


21.5 


21.5 


760 


12.1 


11.8 


11.3 


10.5 


10.0 


9.3 


8.5 


7.9 


7.3 


21.2 


21.1 


21.1 


21.2 


780 


12.8 


12.3 


11.9 


11.4 


10.9 


10.2 


9.6 


9.0 


8.2 


20.4 


20.5 


20.6 


20.7 


800 


12.9 


12.9 


12.5 


12.1 


11.7 


11.2 


10.5 


9.8 


9.2 


19.6 


19.8 


19.9 


20.1 


820 


12.5 


12.7 


12.S 


12.7 


12.2 


11.9 


11.2 


10.7 


10.1 


18.8 


19.0 


19.2 


19.4 


840 


12.0 


12.4 


12.6 


12.8 


12.6 


12 4 


12.2 


11.5 


10.9 


18.1 


18.2 


18.4 


18.6 


860 


11.2 


11.8 


12.3 


12.5 


12.7 


12 5 


12.5 


12.3 


11.7 


17.4 


17.5 


17.6 


17.9 


880 


10.1 


11.0 


11.5 


12.1 


12.3 


12.6 


12.6 


12.4 


12.3 


16.9 


16.9 


16.9 


17.1 


900 


8.8 


9.8 


10.6 


11.3 


11.8 


12.2 


12.4 


12.5 


12.4 


16.3 


16.4 


16.4 


16.5 


920 


7.4 


8.4 


9.3 


10.2 


11.0 


11.5 


12.1 


12.2 


12.3 


16.0 


15.9 


15.9 


16.0 


940 


5.9 


7.1 


8.1 


8.9 


9.9 


10.7 


11.2 


11.7 


12.1 


15.8 


15.7 


15.7 


15.6 i 


960 


4.6 


5.6 


6.7 


7.7 


8.7 


9.4 


10.2 


10.9 


11.4 


15.5 


15.4 


15.3 15.4: 


980 


3.3 


4.2 


5.2 


6.2 


7.3 


8.2 


8.9 


9.9 


10.6 


15.3 


15.2 


15.2 15.lt 


1000 


2.2 
920 


3.0 
930 


3.8 
940 


4.8 
950 


5.8 
950 


6.9 


7.8 
980 


8.7 
990 


9.5 

1000 


15.3 


15.1 


15.0 


15.0 


970 





10 


20 


30 



TABLE XXXII. 



39 



Perturbations p r o>luced by Jupiter. 

Arguments II. and V. 

V. 



1 n. 




5 


: 


j 


91 90 


' 100 110 


120 


is: 


140 


150 







' 






:4~ 


14.6 




• 

14.5 


"... 


1 1 ' 




14.6 


14 : 


14 a 




I 






14.6 


1 ! , 


1 _ _ 


14 2 


14 2 


14.1 


14 1 


14.1 


14.1 


14.: 


14,2 




40 








14 : 




14.1 


13.9 


u a 


11 S 


13.8 


IS 


13 5 










1 


14 


- 


14.1 


13.9 


15 a 


;: : 


Iff 5 




13.5 


i? 4 




S 










14.0 


1 ! ; 


ii • 


is r 


11 . 


i: 2 


13.1 


i 


13.1 




1 






. 




ii a 


1 ■ 


13.6 


1 : ' 


13.4 


13 2 


13.0 


12.2 






12 


13.3 






! : s 


i: " 


' 


13.5 


:; : 


13.3 


13.2 




:•: 5 


12 E 




140 




. 


13.0 


! 




1 2 


ii : 


j : : 


u : 


13.0 


12 


:•; a 


12 : 




160 


:: 3 


12 


12 1 


12.6 




12 B 


12 : 


12 :- 


13.0 


12 : 


12 B 


12 3 


12 z 




IS 




11.1 


11 


11.9 




12 : 


12 '- 


12 5 


12 




:•; a 


12 E 


12.5 




a 


9.5 


10.0 


10.6 


11.0 


11 5 


ii : 


- 


: - 






12 4 


12 


12.3 




22 










IC 1 


i a 


i ! a 


11 : 


11 8 


11.9 


12 : 


IS [ 


12 : 




14 






a 2 


a 9 


. 




i 


10.6 


11.0 


11.3 


11 •: 


11 3 


:: 5 


2 


. 


' 


: 


■ 




8 8 






10.1 


IC 5 


10.9 


11.0 


11 2 


28 


2 


5 5 


6.0 






- : 


2 


' 




9.6 


1( . 




1 


300 




1 


5.1 


5.5 




: : 




i : 


S 1 


a s 


9.1 


r 4 


: \ 






3.9 


LI 


4.6 


5.1 






: 


3 I 






S c 




340 


1 




- 








.-: : 




. 1 


6.6 


: 2 


7 : 


s : 


360 


2 


3.0 


3.1 


3.3 


3.6 


a 8 


--- 




s : 


5 r 


6.1 


e - 


~- 




1 


2.8 


•- a 
- - 


2 3 


2 : 






•: 2 


3.5 


4-1 




5.0 


■ 


400 


' 


: i 


2 : 


2 ; 


2 8 


2 a 






1 


a a 


4-; 


^ 7 


-•; 








3.3 


3 1 


•: a 


2 7 


2 : 


2 : 


a : 


3 2 


3.5 


3 5 




440 


1 




3.9 


S 4 


3.1 








•: a 


3.1 


3.1 


3 2 


3-5 


! 460 








4 


" 


■: 2 


2 


2.8 


- 


2.7 




■; 5 


: 2 


. 






" 




i 


4.1 


3.6 


:1 : 




2 B 


■: a 


2 : 


2.7 






" 


« 




" 


' 


- 


3.9 




" 


3.1 


2.9 


« 




: s 


- 


; : 


6.9 


- 


" 


4.8 


4.2 




3.5 




:- 




ii : 


. 2 


9.1 




- 




- 


■ r ■: 




4-1 


§ ; 


' 


5 


14 


13.0 


11.9 


: a 


9.9 




' 


7.1 




5-8 


5.2 


4,5 


4-1 


' 




. 4 r 


13.6 


12.5 


11.4 


10.4 


: \ 






6.9 


2 


5.5 


T 


600 




16.0 


15.0 


14 


13.1 


12 


n 


ic : 


2 


B 2 




6.7 


" 








16.5 




:.- 


U 


12 i 


u 


1- " 


; 


9.0 


^ : 


7.3 


640 


19.5 


" 


17 


." 


:: 






13.1 


12 •; 


;■ 


_: a 




8.7 i 
11 1 


660 


- 4 


13.7 


18 S 


18.1 




i : 


: r 






12 8 


11.9 


li.O 




21 ■; 


2 5 


19.9 


1 1 


18 5 


:~ 


16.8 


16.0 


1 s 


14 2 


13.5 


12 "' 


:: : 


700 


-: 


2i : 


2 


■;. . 


1 


:■ 


is : 


v i 


15.6 




:■: 9 


il 








2] 


C ■:" 


2C : 


ii i 


15 r 


18.3 


17.5 


:■: 5 


16.1 


15 1 


- 


74 




21 2 


21.1 


-;: - 


2 " 


2 


:;• 4 


15 -i 


15 4 


17.7 


n 2 


16.8 


15 ^ 


> 




a 


21.0 


.. B 






;: ■: 


i; -. 


1 


18 : 


17.9 


17.4 


i: 7 




. i 


. ■ 


2 i 




20.6 


- 


•; 


i: s 




19.1 


18 3 


18.1 


ii 


SuOj 


- : 


- - 


- 


20.3 


•;: 4 


•:■ e 


■:o.i 




II i 


i« : 


19.1 


ie - 


18 2 


82 




19.5 


i« - 


1 


19.9 




... : 


19 3 


ii ; 


19.6 


is a 


18 : 


15 7 


- 




18 ; 




19.0 


I 




: i 


19.4 


1 -_- 


1 


19.4 


19.0 






i 










is ? 


18 


19.0 


:; : 


19.1 


19.0 


18 9 


--:: 




'.' 2 


' 


- 


17.9 


> 


15 •; 






is -: 


18.6 


18.6 






900 




16.6 






M 1 




17.4 


.. . 




it £■ 


18.1 


18 2 


IS 2 




920 




16.0 


16.1 


16.2 


16.4 16.5 


16.7 




:' 


17 2 


17 4 




17.7 




940 


15.6 




15.6 






>; :■ 




16 3 16.5 


16.8 




17.1 




960 






' 


5 


: " 




15 








16.3 






15.1 


150 


15.0 


14.9 


14.9 


14 ? 


14.9 


14.9 


15.0 


IS 2 


15.3 


1-: : 




1000 


15.0 




14.7 




14.6 14.5 






14.5 


14 c 


14.6 




14 S 




30 


_:; 




60 




90 




110 




130 


14: 







40 



TABLE XXXII. 



Perturbations produced by Jupiter. 
Arguments II. and V. 
V. 



II. 




150 


160 


170 


180 


190 


200 


210 
16.2 


1220 
16.3 


230 


240 


250 


260 | 270 


14.8 


15.0 


15.3 


15.5 


15.8 


15.9 


16.7 


17.0 


17.1 


17.3 1 17.5 


20 


14.2 


14.3 


14.6 


H.8 


14.9 


15.2 


15.5 


15.7 


15.9 


16.2 


16.6 


16.8 


17.1 


40 


13.7 


13.7 


13.9 


14.1 


14.3 


14,5 


14.8 


15.0 


15.3 


15.5 


15.8 


16.2 


16.4 


60 


13.3 


13.2 


13.4 


13.5 


13.6 


13.8 


14.1 


14.3 


14.6 


14.8 


15.1 


15.5 


15.§ 


80 


13.1 


13.0 


13.0 


13.0 


13.1 


13.1 


13.3 


13.5 


13.8 


14.1 


14.4 


14.5 


15.1 


100 


12.7 


12.7 


12.7 


12.6 


12.7 


12.6 


12.8 


12.9 


13.1 


13.4 


13.7 


14.0 


14.2 


120 


12.6 


12.5 


12.5 


12.4 


12.3 


12.2 


12.3 


12.3 


12.6 


12.8 


13.0 


13.3 


13.6 


140 


12.6 


12.4 


12.4 


12.3 


12.1 


12.0 


12.0 


12.0 


12.1 


12.1 


12.3 


12.5 


12.8 


: 160 


12.5 


12.3 


12.2 


12.1 


12.1 


11.9 


11.8 


11.8 


11.8 


11.8 


11.9 


12.0 


12.2 


180 


12.5 


12.3 


12.2 


12.1 


11.9 


11.8 


11.7 


11.5 


11.5 


11.5 


11.6 


11.7 


11.8 


200 


12.3 


12.2 


12.2 


12.0 


11.9 


11.7 


11.7 


11.5 


11.4 


11.3 


11.2 


11.3 


11.5 


220 


12.0 


12.0 


12.1 


12.0 


11.8 


11.6 


11.6 


11.5 


11.4 


11.3 


11.2 


11.1 


11.1 


240 


11.8 


11.8 


11.9 


11.9 


11.8 


11.6 


11.5 


11.4 


11.3 


11.2 


11.1 


11.1 


11.0 


260 


11.2 


11.5 


11.6 


11.6 


11.6 


11.5 


11.3 


11.3 


11.3 


11.2 


11.1 


11.0 


10.9 


280 


10.6 


10.8 


11.1 


11.2 


11.2 


11.2 


11.3 


11.3 


11.2 


11.2 


11.1 


11.0 


10.9 


300 


9.9 


10.1 


10.5 


10.8 


10.9 


11.0 


11.1 


11.0 


11.0 


11.0 


11.0 


11.1 


10.9 


320 


8.9 


9.4 


9.7 


10.1 


10.4 


10.5 


10.7 


10.8 


10.8 


10.8 


10.8 


10.8 


10.9 


340 


8.0 


8.5 


9.1 


9.3 


9.6 


9.9 


10.2 


10.3 


10.5 


10.6 


10.6 


10.7 


10.7 


360 


7.1 


7.5 


8.0 


8.4 


8.9 


9.2 


9.5 


9.8 


10.1 


10.3 


10.4 


10.5 


10.5 


380 


5.6 


6.2 


6.8 


7.3 


7.8 


8.3 


8.9 


9.3 


9.7 


10.0 


10.0 


10.1 


10.2 


400 


5.2 


5.6 


6.2 


6.6 


7.0 


7.5 


7.9 


8.4 


8.8 


9.1 


9.4 


9.7 


9.9 


420 


4.3 


4.8 


5.3 


5.8 


6.2 


6.6 


7.1 


7.4 


7.9 


8.4 


8,7 


9.1 


9.4 


440 


3.5 


3.9 


4.4 


4.9 


5,4 


5.7 


6.2 


6.7 


7.1 


7.6 


7.9 


8.4 


8.7 


460 


3.2 


3.3 


3.8 


4.1 


4.5 


4.9 


5.4 


5.7 


6.3 


6.7 


7.2 


7.7 


8.0 


480 


2.7 


2.9 


3.2 


3.6 


3.9 


4.3 


4.7 


5.0 


5.4 


5.9 


6.3 


6.8 


7.3 


500 


2.7 


2.7 


2.9 


3.1 


3.4 


3.6 


4.0 


4.4 


4.8 


5.2 


5.7 


5.9 


6.4 


520 


3.1 


2.8 


2.9 


3.0 


3.1 


3.2 


3.5 


3.8 


4.2 


4,7 


4.9 


5.4 


5.7 


540 


3.5 


3.2 


3.1 


3.0 


3.0 


3.0 


3.3 


3.5 


3.7 


4,1 


4.3 


4.7 


5.1 


560 


4.1 


3.8 


3.6 


3.3 


3.2 


3.2 | 3.2 


3.3 


3.5 


3.7 


4.0 


4,3 


4.5 


580 


5.0 


4.6 


4.2 


4.0 


3.6 


3.5 3.3 


3.2 


3.4 


3 5 


3.7 


4.0 


4.2 


600 


6.0 


5.4 


5.1 


4.6 


4.3 


3.9 3.7 


3.5 


3.5 


3.6 


3.7 


3,8 


4.0 


620 


7.3 


6.6 


6.0 


5.6 


5.1 


4.6 4.2 


4.0 


3.9 


3.8 


3.9 


3.9 


4.0 


640 


8.7 


7.8 


7.3 


6.6 


6.1 


5.5 


5.2 


4,7 


4.4 


4.2 


4.0 


4,0 


4.1 


660 


10.1 


9.3 


8.6 


7.7 


7.2 


6.5 


6.2 


5.9 


5.3 


4.9 


4.6 


4.5 


4.4 


680 


11.6 


10.8 


10.0 


9.3 


8.5 


7.5 


7.3 


6.7 


6.3 


5.8 


5.5 


5.2 


4.9 


700 


13.0 


12.1 


11.5 


10.7 


0.9 


9.0 


8,5 


7.8 


7.4 


6.9 


6.3 


6.0 


5.8 


720 


14.3 


13.5 


12.8 


12.1 


11.3 


10.6 


9.8 


9.1 


8.7 


8.0 


7.6 


7.0 


6.6 


740 


15.7 


14.9 


14.2 


13.4 


12.7 


12.0 


11.2 


10.5 


9.7 


9.3 


8.9 


8.2 


7.7 


760 


16.7 


15.9 


15.5 


14.7 


13.9 


13.3 


12.6 


11.8 


11.2 


10,5 


10.0 


9.5 


9.0 


780 


17.6 


17.0 


16.4 


15.7 


15.1 


14.6 


13.8 


13.2 


12.6 


11.9 


11.2 


10.8 


10,2 


800 


18.2 


17.3 


17.3 


1G.8 


16.2 


16.0 


15.0 


14.3 


13.7 


13.1 


12.6 


12.0 


11.5 


820 


18.7 


18.3 


18.0 


17.6 


17.0 


16.6 


10.0 


15.3 


14.9 


14.3 


13.7 


13.1 


12.6 


S40 


18.9 


18.7 


18.4 


18.2 


17.7 


17.2 


16.8 


16.3 


15.8 


15.3 


14.9 


14.4 


13.8 


860 


18.8 


18.7 


18.6 


18.4 


18.3 


17.9 


17.4 


17.1 


16.7 


16.3 


15.9 


15.4 


15.0 


8S0 


18.7 


18.5 


18.6 


18.5 


18.3 


18.2 


18.0 


17.7 


17.4 


17.1 


16.6 


16.3 


15.9 


900 


18.2 


18.2 


18.3 


18.3 


18.3 


18.1 


18.1 


18.0 


17.8 


17.6 


17.3 


17.0 


16.7 


920 


17.7 


17.9 


18.0 


18.0 


18.1 


18.1 


18.0 


18.0 


18.0 


17.8 


17.7 


17.6 


17.3 


940 


17.1 


17.1 


17.4 


17.6 


17.6 


17.7 


17.8 


17.8 


17.9 


18.0 


17.8 


17.8 


17.7 


960 


16.3 


16.5 


16.8 


16.9 


17.1 


17.2 


17.4 


17.5 


17.6 


17.8 


17.9 


18.0 


17.9 


980 


15.5 


15.7 


16.1 


16.3 


16.5 


16.7 


16.8 


17.0 


17.2 


17.3 


17.6 


17.7 


17.9 


1000 


14,8 


15.0 


15.3 


15.5 


15.8 


15.9 

200 


16.2 
210 


16.3 

220 


16.7 


17.0 


17.1 


17.3 


17.5 


150 


160 


170 


180 


190 


230 


240 


250 


260 


270 



taele ::xxu. 



41 



Perturbations produced by Jupiter. 
Arguments II. and V 



II. 


270 


j 2S0 


] 290 


| 300 


310 


320 


330 


340 


350 


360 


j 370 


380 


390 


., 


// 


,/ 


// 


/, 


,. 


~T\~ 


„ 


,, 





17.5 


17.5 


17.7 


17.8 


17.9 


17.9 


18.0 


18.0 


17.9 


17.7 


17.6 


17.5 


17.5 


20 


17.1 


17.3 


17.5 


17.6 


17.8 


17.8 


18.0 


18.1 


18.1 


18.1 


18.0 


18.0 


18.0 


40 


16.4 


16.8 


16.9 


17.2 


17.6 


17.7 


17.9 


18.1 


18.3 


18.3 


18.4 


18.4 


18.6 


60 


15.8 


16.0 


16.4 


16.7 


16.9 


17.3 


17.6 


17.9 


18.2 


18.3 


18.5 


18.5 


18.7 


SO 


15.1 


15.4 


15.7 


16.1 


16.4 


16.7 


17.0 


17.5 


17.8 


18.0 


18.3 


18.5 


18.8 


100 


14.2 


14.6 


15.1 


15.0 


15.8 


16.1 


16.5 


17.0 


17.2 


17.5 


17.9 


18.3 


18.7 


120 


13.6 


13.7 


14.2 


14.5 


15.0 


15.4 


15.8 


16.2 


16.7 


17.1 


17.3 


17.9 


18.3 


140 


12.8 


13.1 


13.3 


13.7 


14.2 


14.4 


15.1 


15.5 


15.9 


16.3 


16.8 


17.3 


17.7 


160 


12.2 


12.4 


12.6 


12.9 


13.4 


13.8 


14.1 


14.6 


15.2 


15.5 


16.0 


16.5 


17.1 


ISO 


11.8 


11.9 


12.1 


12.3 


12.5 


12.8 


13.3 


13.7 


14.4 


14.7 


15.2 


15.7 


16.3 


200 


11.5 


11.5 


11.6 


11.7 


12.0 


12.1 


12.5 


13.0 


13.4 


13.8 


14.3 


14.7 


15.5 


220 


11.1 


11.1 


11.2 


11.3 


11.6 


11.7 


11.9 


12.3 


12.7 


13.0 


13.5 


14.0 


14.5 


240 


11.0 


10.9 


10.9 


11.0 


11.2 


11.3 


11.5 


11.8 


12.1 


12.3 


12.8 


13.2 


13.8 


260 


10.9 


10.8 


10.8 


10. S 


10.9 


10.9 


11.1 


11.3 


11.4 


11.6 


12.0 


12.3 


13.0 


280 


10.9 


10.8 


10.7 


10.6 


10.7 


10.6 


10.8 


11.0 


11.2 


11.3 


11.5 


11.8 


12.2 


300 


10.9 


10.8 


10.7 


10.6 


10.6 


10.5 


10.6 


10.7 


10.8 


10.9 


11.1 


11.4 


11.8 


320 


10.9 


10.7 


10.7 


10.6 


10.6 


10.5 


10.5 


10.6 


10.7 


10.6 


10.7 


11.0 


11.2 


340 


10.7 


10.7 


10.6 


10.5 


10.5 


10.4 


10.5 


10.5 


10.6 


10.5 


10.6 


10.7 


10.8 


3G0 


10.5 


10.5 


10.5 


10.5 


10.5 


10.4 


10.4 


10.4 


10.4 


10.3 


10.5 


10.6 


10.8 


380 


10.2 


10.3 


10.3 


10.3 


10.4 


10.3 


10.4 


10.4 


10.4 


10.3 


10.3 


10.4 


10.6 


400 


9.9 


10.0 


10.0 


10.2 


10.3 


10.2 


10.2 


10.3 


10.4 


10.3 


10.3 


10.3 


3.0.5 


420 


9.4 


9.6 


9.8 


9.9 


10.1 


10.2 


10.1 


10.2 


10.2 


10.2 


10.3 


103 


10.4 


440 


S.7 


9.0 


9.2 


9.4 


9.7 


9.8 


10.0 


10.1 


10.2 


10.1 


10.1 


10.2 


10.4 


460 


8.0 


S.4 


S.6 


8.8 


9.1 


9.3 


9.6 


9.9 


10.1 


10.0 


10.0 


10.2 


10.3 


480 


7.3 


7.6 


7.9 


8.4 


8.7 


8.9 


9.1 


9.4 


9.6 


9.7 


9.8 


10.0 


10.1 


500 


6.4 


6.9 


7.2 


7.6 


8.0 


8.3 


8.6 


S.9 


9.2 


9.4 


9.5 


9.7 


9.9 


520 


5.7 


6.1 


6.6 


6.9 


7.3 


7.6 


7.9 


8.3 


8.6 


8.9 


9.1 


9.4 


9.7 j 


540 


5.1 


5.4 


5.8 


6.2 


6.7 


7.0 


7.4 


7.7 


8.0 


8.3 


8.6 


8.9 


9.2! 


560 


4.5 


4.9 


5.1 


5.5 


6.0 


6.3 


6.7 


7.2 


7.5 


7.7 


8.0 


S.3 


8.7 


580 


4.2 


4.4 


4,8 


5.0 


5.3 


5.7 


6.1 


6.6 


6.9 


7.1 


7.4 


7.7 


8.1 


600 


4.0 


4.2 


4.3 


4.7 


4.9 


5.2 


5.6 


6.0 


6.3 


6.5 


6.8 


7.2 


7.6 


620 


4.0 


4.0 


4,1 


4.3 


4,7 


4.S 


5.1 


5.5 


5.8 


6 1 


6.4 


6.7 


7.0 


640 


4.1 


4.1 


4.2 


4,2 


4,4 


4.6 


4.8 


5.1 


5.4 


5.6 


5.9 


6.3 


6.6 


660 


4.4 


4,3 


4.3 


4.3 


4.5 


4.5 


4.7 


4.9 


5.1 


5.3 


5.5 


5.8 


6.2 


680 


4.9 


4.9 


4.7 


4.6 


4.7 


4.5 


4.6 


4.8 


5.0 


5.1 


5.3 


5.5 


5.8 


700 


5.8 


5.4 


5.2 


5.1 


5.0 


4,9 


4.9 


4.9 


5.1 


5.2 


5.3 


5.4 


5.6 


720 


6.6 


6.2 


5.9 


5.7 


5.6 


5.5 


5.4 


5.3 


5.3 


5.3 


5.3 


5.4 


5.5 


740 


7.7 


7.2 


6.8 


6.5 


6.4 


6.1 


6.0 


5.9 


5.8 


5.7 


5.6 


5.5 


5.7 


760 


9.0 


8.2 


7.9 


7.5 


7.2 


6.9 


6.7 


6.5 


6.3 


6.1 


5.9 


5.9 


6.0 


780 


10.2 


9.7 


9.1 


8.4 


8.2 


7.7 


7.6 


7.4 


7.2 


6.9 


6.6 


6.5 


6.5 


800 


11.5 


11.0 


10.4 


9.8 


9.4 


8.7 


8.5 


8.3 


8.0 


7.7 


7.6 


7.3 


7.1 


820 


12.6 


12.1 


11.7 


11.2 


10.6 


10.1 


9.7 


9.2 


9.1 


8.6 


8.3 


8.1 


7.9 


840 


13.8 


13.2 


12.8 


12.3 


11.9 


11.3 


10.9 


10.5 


10.2 


9.6 


9.4 


9.1 


8.9 


860 


15.0 


14.4 | 


13.8 


13.5 


13.1 


12.6 


12.1 


11.7 


11.2 


10.7 


10.4 


10.1 


10.0 


880 


15.9 | 


15.4 


15.0 


14.4 


14.2 


13.7 


13.4 


12 9 


12.5 


12.0 


11.5 


113 


11.1 


900 


16.7 


16.4 


15.9 


15.5 


15.2 


14.8 


14.4 


14.1 


13.7 


13.2 


12.8 


12.4 


12.2 


920 i 


17.3 


17.1 


16.8 


16.5 


16.2 


15.7 


15.5 


15.2 


14.8 


14.3 


14.0 


13.6 


13.3 


940 


17.7 ! 


17.5 


17.3 


17.1 


16.9 


1G.6 


16.3 


16.1 


16.0 


15.5 


15.0! 


14.7 


14.5 


960 


17.9 j 17.8 ! 


17.6 


17.5 


17.4 


17.2 


17.0 


16.9 


16.8 


16.4 


16.2 


15.8 


15.6 


980 


17.9 17.8 


17.8 


17.8 


17.8 


17.8 


17.6 


17.5! 


17.3 


17.2 


17.0 


16.8 


16.6 


1000 


17.5 


17.7 


17.7 
290. 


17.8 

300 | 


17.9 


17.9 


18.0 


18.0 
340 


17.9 
350 


17.7 


17.6 


17.5 


17.5 


27C| 


280 


310 


320 | 


330 


360 


370 


380 


390 



42 



TABLE XXXII. 



Perturbations produced by Jupiter. 

Arguments II. and V. 

V. 



II. 


390 


400 


410 


420 


430 


440 


450 


460 


470 


480 


490 


500 


1 51Q 





17.5 


17.1 


17.0 


16.7 


16.5 


16.3 


16.1 


15.8 


15.6 


15.1 


14.6 


14.3 


13.9 


20 


18.0 


18.1 


17.7 


17.5 


17.5 


17.2 


17.1 


16.8 


16.7 


16.3 


16.0 


15.6 


15.3 


40 


18.6 


18.6 


18.5 


18.4 


18.3 


18.1 


18.0 


17.8 


17.6 


17.3 


17.2 


16.8 


16.5 


60 


18.7 


18.9 


18.9 


18.9 


18.9 


18.7 


18.8 


18.6 


18.7 


18.4 


18.1 


17.9 


17.7 


SO 


18.8 


18.9 


19.2 


19.3 


19.4 


19.3 


19.3 


19.3 


19.3 


19.2 


19.2 


18.9 


18.8 


100 


18.7 


18.9 


19.1 


19.4 


19.7 


19.8 


19.8 


19.8 


19.8 


19.8 


19.9 


19.7 


19.7 


120 


18.3 


18.6 


18.9 


19.2 


19.5 


19.8 


20.0 


20.1 


20.3 


20.3 


20 4 


20.4 


20.4 


140 


17.7 


18.2 


18.6 


18.9 


19.2 


19.6 


20.0 


20.3 


20.5 


20.6 


20.7 


20.8 


21.0 


160 


17.1 


17.6 


17.9 


18.5 


19.0 


19.3 


19.8 


20.2 


20.5 


20.6 


20.9 


21.1 


21.2 


ISO 


1G.3 


16.8 


17.3 


17.9 


18.3 


18.8 


19.3 


19.8 


20.3 


20.6 


20.9 


21.1 


21.4 


200 


15.5 


16.0 


16.5 


17.1 


17.7 


18.2 


18.6 


19.1 


19.8 


20.2 


20.7 


21.0 


21.4 


220 


14.5 


15.0 ! 15.6 


16.1 


16.9 


17.4 


18.0 


18.6 


19.0 


19.7 


20.3 


20.7 


21.1 


240 


13.8 


14.2 


14.7 


15.2 


15.9 


16.5 


17.1 


17.7 


18.4 


18.9 


19.5 


20.1 


20.7 


260 


13.0 


13.4 


13.9 


14.4 


15.0 


15.5 


16.3 


16.9 


17.5 


18.0 


18.6 


19.3 


20.0 


280 


12.2 


12.7 


13.0 


13.5 


14.2 


14.7 


15.3 


15.9 


16.7 


17.2 


17.8 


18.4 


19.1 


300 


11.8 


11.9 


12.4 


12.8 


13.3 


13.8 


14.4 


14,9 


15.7 


16.3 


17.0 


17.6 


18.2 


320 


11.2 


11.5 


11.8 


12.2 


12.7 


13.0 


13.6 


14.1 


14.7 


15.3 


16.0 


16.6 


17.4 


340 


10.8 


11.2 


11.4 


11.6 


12.1 


12.4 


12.9 


13.4 


13.9 


14.4 


15.1 


15.7 


16.4 


360 


10.8 


10.8 


11.0 


11.2 


11.6 


11.9 


12.3 


12.6 


13.2 


13.6 


14.2 


14.8 


15.5 


380 


10.6 


10.6 


10.7 


10.9 


11.2 


11.4 


11.9 


12.2 


12.6 


12.9 


13.5 


13.9 


14.5 


400 


10.5 


10.5 


10.6 


10.6 


10.9 


11.1 


11.4 


11.8 


12.2 


12.5 


12.9 


13.3 


13.8 


420 


10.4 


10.4 


10.5 


10.6 


10.7 


10.9 


11.2 


11.3 


11.7 


11.9 


12.4 


12.S 


13.3 


440 


10.4 


10.4 


10.4 


10.5 


10.7 


10.8 


10.9 


11.1 


11.3 


11.6 


11.9 


12.2 


12.7 


460 


10.3 


10.4 


10.4 


10.4 


10.6 


10.6 


10.7 


10.9 


11.2 


11.3 


11.7 


11.9 


12.2 


480 


10.1 


10.2 


10.3 


10.4 


10.6 


10.6 


10.7 


10.8 


11.0 


11.2 


11.4 


11.7 


12.0 


500 


9.9 


10.0 


10.1 


10.2 


10.4 


10.5 


10.7 


10.8 


10.9 


11.0 


11.2 


11.3 


11.7 


520 


9.7 


9.8 


9.8 


10.0 


10.2 


10.3 


10.5 


10.6 


10.9 


10.8 


11.1 


11.3 


11.5 


540 


9.2 


9.4 


9.6 


9.8 


10.0 


10.2 


10.3 


10.4 


10.6 


10.7 


10.9 


11.1 


11.4 


560 


8.7 


8.9 


9.1 


9.3 


9.7 


9.8 


10.1 


10.3 


10.5 


10.6 


10.7 


10.8 


11.2 


580 


8.1 


8.5 


8.7 


8.7 


9.2 


9.4 


9.7 


9.9 


10.2 


10.4 


10.6 


10.7 


10.9 


600 


76 


7.9 


8.2 


8.5 


8.8 


9.0 


9.3 


9.5 


9.8 


10.0 


10.3 


10.5 


10.7 


620 


7.0 


7.3 


7.6 


7.9 


8.2 


8.5 


8.8 


9.0 


9,4 


9.6 


10.0 


10.1 


10.4 


640 


6.6 


6.8 


7.1 


7.4 


7.7 


7.9 


8.2 


8.6 


8.9 


9.1 


9.4 


9.7 


10.1 


660 


6.2 


6.4 


6.6 


6.9 


7.3 


7.6 


7.9 


8.1 


8.3 


8.6 


8.9 


9.2 


9.5 


680 


5.8 


6.1 


6,2 


6.5 


6.8 


7.0 


7.4 


7.6 


7.9 


8.1 


8.4 


8.7 


9.0 


700 


5.6 


5.8 


6.0 


6.2 


6.4 


6.6 


6.9 


7.1 


7.4 


7.6 


7.9 


8.2 


8.5 


720 


5.5 


5.6 


5.7 


5.9 


6.2 


6.3 


6.5 


6.8 


7.1 


7.2 


7.5 


7.7 


8.0 


740 


5.7 


5.7 


5.7 


5.8 


6.0 


6.1 


6.2 


6.4 


6.7 


6.9 


7.1 


7.2 


7.5 


760 


6.0 


6.0 


6.0 


6.0 


6.0 


6.1 


6.2 


6.3 


6.4 


6.5 


6.7 


6.8 


7.1 


780 


6.5 


6.3 


6.2 


6.2 


6.3 


6.3 


6.3 


6.3 


6.4 


6.4 


6.5 


6.7 


6.8 


800 


7.1 


7.0 


6.7 


6.6 


6.7 


6.5 


6.5 


6.4 


6.5 


6.5 


6.5 


6.6 


S.7 


820 


7.9 


7.6 


7.5 


7.3 


7.2 


7.0 


7.0 


6.8 


6.8 


6.7 


6.6 


6.6 


6.7 


840 


8.9 


8.6 


8.3 


8.1 


7.8 


7.7 


7.6 


7.4 


7.3 


7.1 


7.0 


6.8 


6.8 


860 


10.0 


9.7 


9.3 


9.0 


8.7 


8.4 


8.2 


8.1 


7.9 


7.7 


7.6 


7.3 


7.2 


880 


11.1 


10.5 


10.4 


10.0 


9.7 


9.5 


9.2 


8.9 


8.7 


8.4 


8.2 


7.9 


7.7 


900 


12.2 


11.8 


11.5 


11.0 


10.8 


10.5 


10.3 


9.9 


9.7 


9.4 


9.0 


8.8 


8.5 


920 


13.3 


13.0 


12.6 


12.3 


12.1 


11.5 


11.3 


11.0 


10.6 


10.2 


10.1 


9.7 


9.4 


940 


14.5 


14.1 


13.8 


13.5 


13.2 


12.8 


12.5 


11.9 


11.8 


11.3 


11.0 


10.7 


10.4 


960 


15.6 


15.3 


14.9 


14.6 


14.4 


14.0 


13.7 


13.3 


13.0 


12.5 


12.1 


11.8 


11.5 


980 


10.6 


16.3 


16.0 


15.7 


15.6 


15.2 


14.9 


14.6 


14.2 


13.8 


13.6 


12.9 


12.7 


1000 


17.5 


17.1 


17.0 


16.7 


16.5 


16.3 


16.1 


15.8 


15.6 


15.1 


14.6 


14.3 


13.9 




390 


400 


410 


420 


430 


440 


450 


460 


470 


480 


490 


500 


510 



TABLE XXXII. 



43 



Perturbations 'produced by Jupiter. 
Arguments II. and V. 
V. 



II. 




510 ! 520 


530 


540 


550 | 560 570 

— — — 


580 


590 


600 


610 


620 


630 


13.9 13.4 


13.1 


12.7 


12.1 11.8 11.3 


10.8 


10.2 


9.9 


9.4 


8.9 


, 8 - 4 


20 


15.3 


14.9 


14.4 


13.9 


13.5 13.1 12.5 


12.1 


11.5 


11.0 


10.4 


10.0 


9.4 


40 


16.5 


16.3 


15.7 


15.4 


15.0 14.3 13.8 


13.4 


12.8 


12.3 


11.7 


11.1 


10.5 


60 


17.7 


17.3 


17.0 


16.6 


16.1 15.8 15.3 


14.7 


14.3 


13.7 


13.0 


12.4 


11.8 


80- 


18.8 


1S.5 


18.1 


17.9 


17.4 17.1 16.6 


16.2 


15.7 


15.1 


14.5 


13.9 


13.2 


100 


19.7 


19.5 


19.2 


19.0 


18.8! 18.4 17.9 


17.6 


17.0 


16.5 


16.0 


15.2 


14.7 


120 


20.4 


20.3 


20.2 


20.0 


19.7 19.5 19.1 


IS. 8 


18.4 


18.0 


17.3 


16.8 


16.2 


140 


21.0 


21.1 


21.0 


20.8 


20.7120.4 20.2 


19.9 


19.6 


19.3 


18.8 


18.3 


17.7 


160 


21 2 


21.5 


21.5 


21.6 


21.5,21.3 21.2 


21.0 


20.6 


20.4 


20.1 


19.6 


19.1 


ISO 


21.4 


21.6 


21.8 


22.0 


22.0 22.1 


21.9 


21.8 


21.6 


21.4 


21.1 


20.7 


20.3 


2C0 


21.4 


21.7 


21.9 


22.1 


22.3 


22.5 


22.5 


22.5 


22.4 


22.3 


22.1 


21.8 


21.5 


220 


21.1 


21.5 


21.8 


22.2 


22.5 


22.8 


23.1 


23.1 


22.9 


22.8 


22.9 


22.6 


22.5; 


240 


20.7 


21.1 


21.5 


21.9 


22.3 


22.7 


23.0 


23.3 


23.4 


23.5 


23.4 


23.3 


23.2 


260 


20.0 


20.6 


21.0 


21.6 


22.0 


22.4 


22.8 


23.2 


23.5 


23.8 


23.8 


23.S 


23.9 


2S0 


19.1 


19.9 


20.4 


20.9 


21.5 


22.0 


22.4 


23.0 


23.3 


23.7 


24.0 


24.1 


24.1 


300 


18.2 


19.0 


19.6 


20.3 


20.7 


21.3 


21.8 


22.3 


23.0 


23.4 


23.8 


24.1 


24.3 


320 


17.4 


18.9 


IS. 7 


19.4 


20.0 


20.6 


21.1 


21.8 


22.3 


22.9 


23.3 


23.7 


24.2 


340 


16.4 


17.0 


17.6 


18.5 


19.2 


19.9 


20.4 


21.1 


21.6 


22.2 


22.8 


23.3 


23.7 


3G0 


15.5 


16.2 


16.7 


17.4 


18.2 


18.9 


19.5 


20.1 


20.8 


21.5 


22.0 


22.6 


23.2 


3S0 


14.5 


15.2 


15.9 


16.6 


17.1 


17.9 


18.6 


19.3 


19. S 


20.5 


21.1 


21.8 


22.5 


400 


13.8 


14.4 


14.9 


15.6 


16.2 


16.8 


17.6 


18.4 


19.1 


19.7 


20.3 


20.9 


21.5 


420 


13.3 


13.7 


14.2 


14.8 


15.3 


16.0 


16.5 


17.4 


18.0 


18.7 


19.4 


20.0 


20.6 


440 


12.7 


13.1 


13.6 


14.1 


14.6 


15.2 


15.7 


16.4 


17.1 


17.8 


18.4 


18.9 


19.6 


460 


12.2 


12.7 


13.0 


13.5 


13.9 


14.4 


15.0 


15.6 


16.1 


16.9 


17.5 


18.2 


18.7 


480 


12.0 


12.2 


12.5 


13.0 


13.4 


13.9 


14.3 


14.8 


15.3 


15.9 


16.6 


17.3 


17.9 


500 ( 11.7 


12.0 


12.2 


12.6 


12.9 


13.3 


13.8 


14.3 


14.7 


15.2 


15.7 


16.4 


16.9 


520 11.5 


11.9 


12.0 


12.3 


12.6 


13.0 


13.2 


13.8 


14.2 


14.7 


15.1 


15.5 


16.2 


540 11.4 


11.6 


11.9 


12.2 


12.4 


12.7 


12.9 


13.3 


13.7 


14.2 


14.6 


15.0 


15.4 


560 11-2 


11.4 


11.5 


11.9 


12.1 


12.4 


12.7 


13.1 


13.4 


13.8 


14.1 


14.5 


14.9 


5S0 10.9 


11.2 


11.4 


11.6 


11.9 


12.2 


12.4 


12.8 


13.1 


13.5 


13.8 


14.2 


14.5 


600 10.7 


10.8 


11.1 


11.5 


11.7 


12.0 


12.2 


12.5 


12.8 


13.1 


13.4 


13.8 


14.2 


620 10.4 


10.7 


10.7 


11.1 


11.4 


11.6 


12.0 


12.3 


12.5 


12.9 


13.1 


13.4 


13.8 


640 10.1 


10.4 


10.6 


10.7 


11.0 


11.3 


11.6 


12.0 


12.3 


12.6 


12.9 


13.2 


13.5 


660 9.5 


9.9 


10.2 


10.5 


10.6 


11.0 


11.3 


11.6 


11.9 


12.3 


12.6 


12.9 


13.2 


680 


9.0 


9.3 


9.6 


10.0 


10.3 


10.5 


10.8 


11.3 


11.5 


11.9 


12.2 


12.4 


12.8 


700 


8.5 


8.9 


9.1 


9.5 


9.8 


10.1 


10.3 


10.7 


11.1 


11.4 


11.8 


12.1 


12.4 


720 


S.O 


8.3 


8.5 


9.0 


9.2 


9.6 


9.9 


10.2 


10.5 


10.9 


11.3 


11.7 


12.0 


740 


7.5 


7.8 


8.0 


8.3 


8.6 


9.0 


9.3 


9.7 


9.9 


10.4 


10.8 


11.1 


11.5 


760 


7.1 


7.3 


7.5 


7.9 


8.1 


8.4 


8.6 


9.1 


9.4 


9.7 


10.1 


10.5 


10.9 


780 


6.8 


7.0 


7.1 


7.3 


7.6 


7.9 


8.1 


8.5 


8.8 


9.2 


9.4 


9.8 


10.2 


800 


6.7 


6.8 


6.8 


7.0 


7.1 


7.3 


7.5 


7.8 


8.2 


8.5 


8.8 


9.1 


9.5 


820 


6.7 


6.8 


6.6 


6.8 


6.9 


7.0 


7.1 


7.4 


7.6 


7.9 


8.1 


8.4 


8.7 


840 


6.8 


6.8 


6.8 


6.8 


6.8 


6.9 


6.9 


7.1 


7.2 


7.4 


7.6 


7.9 


8.1 


860 


7.2 


7.1 


7.1 


7.0 


6.9 


6.9 


6.8 


6.8 


6.9 


7.1 


7.2 


7.3 


7.6 


8S0 


7.7 


7.5 


7.4 


7.3 


7.1 


7.0 


6.8 


6.8 


6.7 


6.8 


6.8 


7.0 


7.2 


900 


8.5 


8.2 


7.9 


7.7 


7.5 


7.3 


7.2 


7.1 


6.9 


6.9 


6.8 


6.8 


6.8 


920 


9.4 


9.2 


S.7 


8.4 


8.1 


7.9 


7.6 


7.4 


7.1 


7.0 


6.9 


6.8 


6.7 


940 


10.4 


10.0 


9.7 


9.4 


8.9 


8.6 


8.3 


8.1 


7.7 


7.4 


7.1 


6.9 


6.7 


960 


11.5 


11.2 


10.7 


10.4 


9.8 


9.5 


9.1 


8.8 


8.5 


8.1 


7.7 


7.4 


7.1 


980 


12.7 


12.3 


11.8 


11.5 


11.1 


10.6 


10.0 


9.7 


9.2 


S.9 


8.5 


8.1 


7.7 


1000 


13.9 


13.4 
520 


13.1 


12.7 


12.1 
550 


11.8 


11.3 


10.8 
580 


10.2 


9.9 


9.4 


620 [ 


8.4 
630 


510 


530 


540 


560 


570 


5f;o 


600 


610 



44 



TABLE XXXII. 



Perturbations produced by Jupiter. 

Arguments II. and V". 

V. 



[IL 


630 


640 


650 


660 


670 


680 


690 


700 


710 


720 730 


740 


750 


i 




8.4 


8.0 


7.7 


7.3 


6.9 


6.7 


6.5 


6.5 


6.3 


6.2 6.2 


6.4 


6.5 


20 


9.4 


9.0 


8.4 


8.0 


7.5 


7.1 


6.9 


6.7 


6.4 


6.3 


6.0 


6.1 


6.1 


40 


10.5 


10.1 


9.4 


8.9 


8.3 


7.8 


7.4 


7.0 


6.6 


6.4 


6.2 


5.9 


5.8 


60 


11.8 


11.3 


10.6 


10.1 


9.3 


8.7 


8.2 


7.7 


7.2 


6.8 


6.4 


6.2 


5.8 


80 


13.2 


12.7 


12.0 


11.3 


10.5 


9.9 


9.2 


8.7 


8.1 


7.6 


7.1 


6.6 


6.2 


100 


14.7 


14.1 


13.4 


12.8 


12.0 


11.3 


10.6 


9.9 


9.1 


8.5 


7.9 


7.3 


6.8 


120 


16.2 


15.4 


14.9 


14.2 


13.4 


12.7 


12.0 


11.3 


10.4 


9.8 


8.9 


8.2 


7.6 


140 


17.7 


17.2 


16.4 


15.6 


14.9 


14.2 


13.4 


12.7 


11.9 


11.1 


10.2 


9.6 


8.8 


160 


19.1 


18.6 


17.9 


17.3 


16.6 


15.7 


15.0 


14.2 


13.3 


12.6 


11.7 


10.9 


10.0 


180 


20.3 


19.9 


19.4 


1S.8 


18.0 


17.3 


16.7 


15.8 


15.0 


14.1 


13.2 


12.4 


11.5 


200 


21.5 


21.2 


20.8 


20.2 


19.3 


18.9 


18.1 


17.5 


16.6 


15.7 


14.9 


14.0 


13.1 


220 


22.5 


22.3 


21.9 


21.5 


21.0 


20.3 


19.7 


19.0 


18.2 


17.5 


16.6 


15.5 


14.7 i 


240 


23.2 


23.0 


22.9 


22.5 


22.0 


21.6 


21.1 


20.5 


19.8 


19.1 


18.2 


17.3 


16.41 


260 


23.9 


23.8 


23.7 


23.5 


23.1 


22.7 


22.3 


21.8 


21.2 


20.6 


19.8 


19.1 


18.1 


2S0 


24.1 


24.3 


24.2 


24.2 


24.0 


23.7 


23.5 


23.1 


22.4 


21.8 


21.2 


20.5 


19.8 


300 


24.3 


24.5 


24.6 


24.6 


24.5 


24.4 


24.2 


23.9 


23.6 


23.1 


22.5 


21.9 


21.2 


320 


24.2 


24.5 


24.7 


24.9 


24.8 


24.8 


24.8 


24.7 


24.4 


24.1 


23.7 


23.1 


22.5 


340 


23.7 


24.2 


24.5 


24.7 


25.0 


25.2 


25.1 


25.0 


25.0 


24.9 


24.6 


24.1 


23.7 


360 


23.2 


23.7 


24.2 


24.5 


24.7 


25.0 


25.1 


25.3 


25.4 


25.3 


25.1 


24.9 


24.5 


380 


22.5 


23.1 


23.6 


24.1 


24.4 


24.7 


25.1 


25.2 


25.4 


25.5 


25.4 


25.3 


25.2 


400 


21.5 


22.3 


22.8 


23.4 


23.9 


24.3 


24.7 


25.1 


25.2 


25.4 


25.6 


25.6 


25.5 


420 


20.6 


21.3 


22.0 


22.6 


23.1 


23.6 


24.1 


24.5 


25.0 


25.2 


25.4 


25.6 


25.7 


440 


19.6 


20.3 


21.0 


21.8 


22.3 


22.9 


23.4 


23.9 


24.3 24.8 


25.0 


25.2 


25.6 


460 


18.7 


19.4 


20.1 


20.7 


21.3 


21.9 


22.6 


23.3 


23.6 


24.1 


24.6 


24.8 


25.1 


480 


17.9 


18.5 


19.1 


19.7 


20.3 


21.0 


21.6 


22.2 


22.8 


23.3 


23.8 


24.3 


24.6 


500 


16.9 


17.6 


18.2 


18.8 


19.3 


19.9 


20.7 


21.4 


21.9 


22.5 


22.9 


23.4 


23.9 


520 


16.2 


16.8 


17.3 


17.9 


18.4 


19.0 


19.7 


20.4 


21.0 


21.6 


21.1 


22.6 


23.0 


540 


15.4 


16.1 


16.6 


17.2 


17.5 


18.1 


18.7 


19.3 


19.9 


20.5 


21.2 


22.7 


22.2 


560 


14.9 


15.4 


16.0 


16.5 


16.9 


17.3 


17.9 


18.4 


18.9 


19.6 


20.1 


20.7 


21.3 


580 


14.5 


15.0 


15.3 


15.9 


16.3 


16.7 


17.1 


17.6 


18.1 


18.7 


19.3 


19.8 


20.3 


600 


14.2 


14.6 


14.9 


15.3 


15.8 16.3 


16.6 


17.0 


17.4 


17.9 


18.3 


18.9 


19.4 


620 


13.8 


14.2 


14.6 


14.9 


15.1 


15.7 


16.2 


16.6 


16.9 


17.3 


17.6 


18.0 


18.5 


640 


13.5 


14.0 


14.2 


14.6 


14.8 


15.1 


15.6 


16.1 


16.5 


16.8 


17.1 


17.5 


17.9 


660 


13.2 


13.5 


13.9 


14.3 


14.6 


14.9 


15.2 


15.6 


15.9 


16.4 


16.6 


17.0 


173 


680 


12.8 


13.2 


13.5 


13.9 


14.2 


14.5 


14.9 


15.2 


15.6 


16.0 


16.2 


16.5 


16.8 


700 


12.4 


12.9 


13.3 


13.5 


13.8 


14.2 


14.5 


14,9 


15.1 


15.6 


15.9 


16.2 


16.4 


720 


12.0 


12.4 


12. S 


13.2 


13 5 


13.8 


14.2 


14,5 


14.8 


15.1 


15.5 


15.8 


16.1 


740 


11.5 


11.9 


12.2 


12.6 


12.9 


13.3 


13.8 


14.2 


14.5 


14.8 


15.1 


15.4 


15.7 


760 


10.9 


11.4 


11.8 


12.2 


12.4 


12.8 


13.2 


13.7 


14.1 


14.5 


14.7 


15.0 


15.4 


780 


10.2 


10.6 


11.2 


11.6 


11.9 


12.4 


12.8 


13.2 


13.5 


13.9 


14.3 


14.6 


14.9 


800 


9.5 


10.0 


10.3 


10.9 


11.3 


11.6 


12.1 


12.6 


12.9 


13.4 


13.8 


14.2 


14.5 


820 


8.7 


9.3 


9.7 


10.0 


10.5 


10.9 


11.4 


11.9 


12.3 


12.8 


13.2 


13.6 


14.0 


840 


8.1 


8.4 


8.8 


9.3 


9.6 


10.1 


10.6 


11.1 


11.6 


12.1 


12.5 


13.0 


13.4 


860 


7.6 


7.9 


8.1 


8.5 


8.8 


9.2 


9.7 


10.2 


10.7 


11.2 


11.7 


12.1 


12.6 


880 


7.2 


7.4 


7.6 


7.8 


8.1 


8.5 


8.8 


9.4 


9.8 


10.2 


10.7 


11.2 


11.8 


900 


6.8 


7.0 


7.1 


7.3 


7.4 


7.8 


8.2 


8.5 


8.9 


9.4 


9.8 


10.3 


10.8 


920 


6.7 


6.8 


6.8 


6.9 


7.0 


7.0 


7.4 


7.8 


8.1 


8.6 


8.9 


9.4 


9.9 


940 


6.7 


6.7 


6.7 


6.8 


6.7 


6.8 


6.8 


7.1 


7.4 


7.7 


8.1 


8.4 


8.9: 


960 


7.1 


7.0 


6.8 


6.7 


6.5 


6.5 


6.6 


6.7 


6.8 


7.1 


7.3 


7.7 


8.0 


980 


7.7 


7.4 


7.1 


6.9 


6.6 


6.5 


6.4 


6.4 


6.3 


6.5 


6.8 


6.9 


7.3 


1000 


8.4 
633 


8.0 


7.7 


7.3 


6.9 


6.7 


6.5 

690 


6.5 

700 


6.3 

710 


6.2 

720 


6.2 


6.4 


6.5 


646 


650 


660 


670 


680 


730 


740 


750 



TABLE XXXII. 



46 



Perturbations produced by Jupiter. 

Arguments II. and V. 

V. 



| II. 750 

1 


760 


7 70 


750 


790 


SOO 


810 


B20 


1 S30 


S40 

1 


S50 


: S60 ; S70 


6.5 


3 


l // 
7 2 


T* 


S.O 


S.4 


8.8 


9.5 


10.1 


10.5 


11.0 


11.6 


12.4 


20 G.l 


6.2 


6.5 


6.7 


7.0 


7.4 


7.9 


8.4 


9.0 


9.5 


10.0 


10.6 


11.1 


40 




5.9 


5.9 


6.2 


6.4 


6.6 


6.9 


7.4 




8.2 


8.S 


9.5 


10.0 


60 


5.8 


5.7 


5.7 


5.7 


5.9 


6.1 


6.2 


6.5 


6.9 


7.2 


7.7 


S.3 


1 8.8 


80 


6.2 


5.S 


5.7 


1 5.6 


5.4 


5.6 


5.7 


1 5.9 


6.1 


6.3 


6.7 


7.3 


7.8 


100 


6.8 


6.3 


5.9 


5.6 


5.5 


5.3 


5.3 


5.4 


5.4 


5.6 


5.9 


6.3 


6.8 


120 


7.6 


7.4 


6.5 


6.0 


5.7 


5.5 


5.1 


5.2 


5.1 


5.1 


5.2 


5.5 


5.8 


140 


B.8 


8.1 


7.4 


6.8 


6.2 


5.S 


5.4 


5.2 


5.0 


4.9 


4.8 


5.0 


5.1 


160 


10.0 


9.3 


8.5 


7.8 


7.2 


| 6.5 


5.9 


5.5 


5.1 


5.9 


4.7 


4.7 


4.7 


ISO 


11.5 


10.6 


9.7 


9.0 


8-2 


7.5 


6.9 


6.3 


5.8 


5.2 


4.5 


4.7 


4.5 


200 


13.1 


12.2 


11.2 


10.4 


9.5 


S.S 


7.9 


7.1 


6.5 


5.9 


5.3 


5.0 


4.7 


220 


14 7 


13.8 


12.9 


12.0 


11.1 


10.2 


9.3 


8.4 


7.5 


6.7 


6.1 


5.5 


5.2 




16.4 


15.3 


14.5 


13.6 


12.6 


11.7 


10.7 


9.8 


8.8 


7.9 


7.0 


6.5 


5.9 


260 


18.1 


17.2 


16.3 


15.3 


14.3 


13.3 


12.2 


11.4 


10.4 


9.4 


8.3 


7.7 


6.9 


2S0 


19.8 


18.9 




17.0 


16.1 


15.0 




13.0 


11.9 


10.9 


9.9 


8.9 


8.0 


300 


21.2 


CJ.4 


19.6 


18.7 




16. S 


15.8 


14.7 


13.7 


12.6 


11.5 


10.5 


9.4 


320 


22.5 


21.9 


21.2 


20.4 


19.4 


18.5 


17.4 


16.5 


15.5 


14.2 


13.2 


12.3 


11.2 


340 


23.7 


23.0 


22.4 


21.8 


21.1 


20.2 


19.2 


IS 3 


17.1 


16.1 


15.0 


13.9 


12.9 


360 


24.5 


34.0 


23.6 


23.0 


22.4 


21.6 


20.8 


19.9 


18.9 


17.9 


16.S 


15.9 


14.7 


380 


25.2 


24.9 


24.5 


24.0 


•23.5 


22.8 


22.1 


21.4 


20.5 


19.5 


18.5 


17.6 


16.5 


400 


25.5 


25.4 


25.1 


24.S 


24.5 


23.9 


23.4 


22.7 


21.9 


21.0 


20.1 


19.2 


18.2 


420 


25.7 


25.6 


25.5 


25.3 


25.0 


24.5 


24.2 


23.7 


23.2 


22.3 


21.5 


20.7 


19 8 


440 


25.G 


25.6 


25.7 


25.7 


25.5 


25.3 


24.9 


24.6 


24.1 


23.4 


22.7 


22.0 


21.2 


460 


25.1 


25.3 


25.5 


25.6 


25.8 


25.7 


25.4 


25.2 


24,8 


24.3 


23.7 


23.1 


22.5 


480 


24.6 


24.9 


25.2 


25.4 


25.6 


25.6 


25.5 


25.4 


25.2 


24.9 


24.5 


24.1 


23.5 


500 


23.9 


24.2 


24.7 


25.0 


25.3 


25.4 


25.5 


25.5 


25.4 


25.2 


24.9 


24.7 


24.3 


520 


23.0 


23.6 


23.9 


24.3 


24.7 


24.9 


25.2 


25.4 


25.4 


25.3 


25.2 


25.1 


24.8 


540 


22.2 


22.6 


23.2 


23.6 


24.0 


24.4 




24.9 


25.1 


25.0 


25.1 


25.1 


25.0 


560 


21.3 


21.7 


22.2 


22.8 


23.2 


23.7 


24.0 


24.3 


24.6 


24.7 


24.S 


24.9 


24.9 


5S0 






21.3 


21. S 


22.3 


22.7 


23.2 


23.7 


23.9 


24.1 


24.4 


24.6 


24.7 


600 


19.4 


19.9 


20.4 


20.S 


21.4 


21.9 


22.2 


22.7 


23.1 


23.4 


23.7 


24.1 


24.3 


620 


18.5 


19.0 


19.5 


20.1 


20.5 


20.9 


21.4 


21.8 


22.2 


22.6 


22.9 


23.3 


23.6 


640 


17.9 


IS. 3 


18.7 


19.2 


19.7 


20.1 


20.5 


22.0 


21.3 


21.7 


22.1 


22.5 


22'.8 


660 


17.3 


17.6 


18.1 


1S.5 


IS. 9 


19.4 


19.6 


20.1 


20.5 


20.7 


21.2 


2L7 


22.0 


680 


16.S 


17.1 


17.4 


17.8 


18.2 


18.6 


18.9 


19.4 


19.7 


20.1 


20.4 


20 7 


21.2 


700 


16.4 


16.7 


16.9 


17.3 


17.7 


18.0 


18.3 


15.7 


18.9 


19.2 


19.6 


20.0 


20.3 


720 


16.1 


16.3 


16.5 


16.9 


17.2 


17.6 


17.8 


18.0 


18.3 


18.5 


18.71 


19 2 


19.5 


740 


15.7 


16.0 


16.2 


16.5 


16.7 


17.0 


17.3 


17.6 


17.8 


17.9 | 


18.1 


18 5 


18.8 


760 


15.4 


15.7 


16.0 


16.1 


16.4 


16.6 


16.7 


17.2 


17.4 


17.4! 


17.8! 


18 


18.2 


780 


14.9 


15.3 


15.6 


15.9 


16.1 


16.3 


16.5 


16.7 


16.9 


17.1 


17.3 j 


17.6 


17.7 


800 


14.5 


14 7 


15.2 


15.5 


15.8 


15.9 


16.2 


16.5 


16.6 


16.S 


16.9 


17.1 


17.3 


820 


14.0 


14.4 


14.7 


15.1 


15.4 


15.7 


15.8 


16.1 


16.3 


16.4 


16.6 


16.9 


17.0 j 


840 


13.4 


13.7 


14.1 


14.5 


15.1 


15.4 


15.4 


15.8 


15.9J 


16.1 


16.2 ! 


16.6 


16.7 ; 


860 


12.6 


13.1 


13.5 


13.9 


14.3 


14.8 


15.2 


15.5 


15.6 J 


15.8 


16.0 ! 


16.3 


16.4 i 


880 


11.8 


12.3 


12.8 


13.3 


13.7 


14.1 


14.5 


15.0 


15.3 


15.4 


15.6, 


15.9 


16.1 


900 


10.8 


11.3 


11.9 


12.4 


13.0 


13.4 


13.7 


14.2 


14.7 


15.0 


15.2 


15.5 


15.7 


920 


9.9 


10.3 


10.8 


11.4 


12.0 


12.5 


12.9 


13.4 


14.0 1 


14.31 


14.71 


15.0 


15.3! 


940 


8.9 


9.4 


9.9 


10.4 


11.0 


11.6 


12.1 


12.5 


13.0! 


13.6 


13.9 


14.4 


14.7 ' 


960 


8.0 


8.3 


8.8 


9.4 


10.0 


10.6 


11.1 


11.7! 


12.2 


12.5 i 


13.1 1 


13.7 


14.1 


980 


7.3 


7.6 


7.9 


8.4 | 


S.9 


9.5 


9.9 


10.5 


11.1 


11.6) 


12.1 


12.8 


13.3 


1000 


6.5 


6.8 
760 i 


7.2 
770 


7.5 
780 


790 


8.4 

SOO 


8.8 
S10 


9.5 

820 \ 


10.0 


10.5 
840 


11.0 
850 | 


11.6 
860 


12.41 




750 


830 \ 


870 



46 



TABLE XXXII. 



Perturbations produced by Jupiter. 

Arguments II. and V. 

V. 



II. 


870 


880 


890 
13.2 


900 


910 


920 


930 

14.4 


940 


950 


960 
15.1 


970 
15.1 


980 


990 


1000 


i •' 
\ 12.4 


12.9 


13.6 


13.9 14.2 


14.8 


15.0 


15.2 


15.2 


15.3 


20 11. 1 


11.7 


12.2 


12.7 


13.2 13.6 


13.8 


14.1 


14.4 


14.7 


14.8 


15.0 


14.9 


14.9 


40 10.0 


10.5 


11.1 


11.7 


12.3 12.6 


13.0 


13.4 


13.7 


14.1 


14.3 


14.6 


14.7 


14.7 


60 


8.8 


9.4 


9.9 


10.6 


11.2 11.8 


12.1 


12.6 


12.9 


13.3 


13.6 


13.9 


14.2 


14.4 


SO 


7.8 


8.3 


8.7 


9.3 


10.0 10.5 


11.1 


11.6 


12.1 


12.5 


12.8 


13.2 


13.5 


13.8 


100 


6.8 


7.2 


7.6 


8.1 


8.6 9.4 


9.9 


105 


10.9 


11.4 


12.0 


12.4 


12.8 


13.2 


120 


5.8 


6.1 


6.6 


7.1 


7.6 8.1 


8.7 


9.4 


9.9 


10.4 


10.S 


11.4 


11.8 


12.3 


140 


5.1 


5.3 


5.6 


6.0 


6.5 7.0 


7.5 


8.2 


8.7 


9.3 


9.7 


10.3 


10.8 


11.3 


160 


4.7 


4.8 


4.8 


5.2 


5.6 5.9 


6.3 


6.8 


7.4 


8.0 


8.6 


9.2 


9.7 


10.2 


180 


4.5 


4.5 


4.4 


4.5 


4.8 


5.1 


5.4 


5.8 


6.2 


6.9 


7.4 


8.0 


3.4 


9.1 


200 


4.7 


4.5 


4.2 


4.2 


4.2 


4.4 


4.6 


5.0 


5.3 


5.7 


6.3 


6.9 


7.4 


7.8 


220 


5.2 


4.7 


4.3 


4.2 


4.1 


4.1 


4.0 


4.3 


4.5 


4,8 


5.1 


5.7 


6.2 


6.8 


240 


5.9 


5.3 


4.7 


4.3 


4.1 


4.0 


3.8 


3.9 


4.0 


4.2 


4.3 


4.7 


5.2 


5.7 


260 


6.9 


6.1 


5.4 


4.9 


4.4 


4.1 


3.8 


3.7 


3.6 


3.7 


3.8 


4.1 


4.3 


4.9 


280 


8.0 


7.2 


6.3 


5.7 


5.2 


4.6 


4.1 


3.8 


3.5 


3.5 


3.5 


3.6 


3:7 


3.9 


300 


9.4 


8.5 


7.5 


6.8 


6.1 


5.4 


4.7 


4.3 


3.9 


3.6 


3.3 


3.3 


3.3 


3.4 


320 


11.2 


10.1 


9.1 


8.1 


7.3 


6.5 


5.7 


5.0 


4.4 


4.0 


3.6 


3.4 


3.2 


3.2 


340 


12.9 


11.8 


10.7 


9.6 


8.7 


7.7 


6.8 


6.0 


5.2 


4.6 


4.1 


3.7 


3.4 


3.2 


360 


14,7 


13.4 


12.3 


11.1 


10.1 


9.2 


8.3 


7.4 


6.4 


5.7 


4.9 


4.3 


3.8 


3.5 


380 


16.5 


15.4 


14.2 


13.0 


11.8 


10.8 


9.7 


8.7 


7.8 


6.9 


6.1 


5.4 


4.6 


4.1 


400 


18.2 


17.2 


16.0 


14.9 


13.8 


12.4 


11.4 


10.4 


9.3 


8.3 


7.3 


6.4 


5.6 


5.0 


420 


19.8 


18.8 


17.7 


16.7 


15.5 


14.4 


13.1 


11.9 


10.9 


9.8 


8.8 


8.0 


6.9 


6.1 


440 


21.2 


20.3 


19.3 


18.3 


17.3 


16.2 


14.9 


13.8 


12.7 


11.5 


10.5 


9.5 


8.4 


7.5 


460 


22.5 


21.6 


20.6 


19.7 


18.9 


17.9 


16.7 


15.6 


14.3 


13.3 


12.2 


10.9 


10.0 


9.0 


480 


23.5 


22.7 


22.0 


21.1 


20.2 


1.9.3 


18.2 


17.3 


16.2 


15.0 


13.8 


128 


11.6 


10.5 


500 


24.3 


23.8 


23.0 


22.3 


21.6 


20.7 


19.7 


18.8 


17.8 


16.7 


15.4 


14.5 


13.4 


12.3 


520 


24.8 


24.3 


23.7 


23.2 


22.7 


21.9 


21.1 


20.2 


19.2 


18.3 


17.2 


16.1 


15.0 


14.0 


540 


25.0 


24.8 


24.3 


23.9 


23.4 


22.8 


22.1 


21.3 


20.6 


19.7 


18.7 


17.6 


16.6 


15.6 


560 


24.9 


24.8 


24.7 


244 


24.0 


23.6 


22.9 


22.4 


21.0 


20.8 


20.0 


19.1 


18.2 


17.1 


580 


24.7 


24.7 


24.6 


24.5 


24.3 


23.9 


23.5 


23.1 


22.5 


21.9 


21.1 


20.3 


19.5 


18.6 


600 


24.3 


24.3 


24.3 


24.3 


24.3 


24.1 


23.8. 


23.5 


23.0 


22.5 


22.0 


21.4 


20.6 


19 8 


620 


23.6 


23.7 


23.9 


24.0 


24.1 


24.1 


23.9 


23.7 


23.4 


23.1 


22.6 


22.1 


21.4 


20.8 


640 


22.8 


23.1 


23.2 


23.4 


23.6 


23.7 


23.8 


23.7 


23.5 


23.2 


22.9 


22.6 


22.1 


21.6 


660 


22.0 


22.3 


22.5 


22.8 


23.0 


23.2 


23.2 


23.3 


23.2 


23.1 


23.0 


22.8 


22.5 


22.1 


680 


21.2 


21.5 


21.7 


22.0 


22.3 


22.5 


22.6 


22.8 


22.9 


22.9 


22.8 


22.7 


22.7 


22.3 


700 


20.3 


20.7 


20.9 


21.2 


21.5 


21.7 


21.9 


22.2 


22.3 


22.5 


22.5 


22.5 


22.4 


22.2 J 


720 


19.5 


19.8 


20.1 


20.4 


20.8 


21.1 


21.2 


21.4 


21.6 


21.8 


21.9 


22.0 


22.0 


22.0 


740 


18.8 


19.0 


19.2 


19.6 


19.9 


20.2 


20.5 


20.7 


20.9 


21.1 


21.2 


21.5 


21.5 


21.6 


760 


18.2 


18.5 


18.4 


18.8 


19.1 


1.9.4 


19.6 


19.9 


20.1 


20.3 


20.5 


20.8 


21.0 


21.2 


780 


17.7 


17.8 


18.0 


18.1 


18.4 


18.7 


18.8 


19.1 


19.3 


19.5 


19.7 


20.0 


20.2 


20.4 


800 


17.3 


17.4 


17.4 


17.7 


17.9 


18.0 


18.1 


18.4 


18.6 


18.9 


18.9 


19.1 


19.4 


19.6 


820 


17.0 


17.2 


17.2 


17.2 


17.4 


17.4 


17.6 


17.8 


17.8 


18.1 


18.3 


18.5 


18.6 


18.8 


840 


16.7 


16.8 


16.8 


16.9 


17.2 


17.2 


17.1 


17.1 


17.3 


17.4 


17.5 


17.8 


17.9 


18.1 


860 


16.4 


16.5 


16.5 


16.6 


16.6 


16.7 


16.8 


16.9 


16.9 


17.0 


17.0 


17.1 


17.2 


17.4 


880 


16.1 


16.3 


16.3 


16.5 


16.5 


16.5 


16.6 


16.6 


16.6 


16.6 


16.6 


16.7 


16.7 


16.9 


900 


15.7 


15.9 


16.1 


16.2 


16.3 


16.4 


16.3 


16.3 


16.2 


16.2 


16.2 


16.3 


16.3 


16.3 


920 


15.3 


15.5 


15.6 


15.9 


16.0 


16.1 


16.1 


16.1 


16.0 


16.1 


16.1 


16.1 


16.0 


16.0 


940 


14.7 


15.9 


15.2 


15.4 


15.7 


15.8 


15.8 


16.0 


15.9 


15.9 


15.9 


15.8 


15.7 


15.8 


960 


14.1 


14.3 


14.5 


14.8 


15.2 


15.5 


15.5 


15.7 


15.7 


15.7 


15.6 


15.6 


15.5 


15.5 


980 


13.3 


12.7 


13.9 


14.2 


14.5 


14.8 


15.1 


15.3 


15.4 


15.5 


15.4 


15.4 


15.4 


15.3 


1000 


12.4 


12.9 


13.2 


13.6 


13.9 

910 , 


14.2 


14.4 


14.8 


15.0 


15.1 


15.1 


15.2 


15.2 


15.3 




870 


880 


890 


900 


920 


930 


940 


950 


960 


970 


980 


990 


1000 



TABLE XXXIII. 
Perturbations pioduced by Saturn. 

Arguments II and VII. 
VII. 



47 



II 





100 


200 


300 


400 


500 


600 


700 


800 


900 


100O 
1.2 





1.2 


1.5 


1.4 


1.0 


0.7 


0.6 


0.5 


0.5 


0.4 


0.8 


100 


0.9 


1.2 


1.3 


1.1 


0.9 


0.8 


0.7 


0.7 


0.6 


0.7 


0.9 


200 


0.7 


0.9 


1.0 


1.1 


1.0 


0.9 


0.8 


0.8 


0.9 


0.8 


0.7 


300 


0.9 


0.8 


0.7 


O.S 


0.9 


1.0 


1.0 


1.0 


1.0 


1.0 


0.9 


400 


1.0 


0.9 


0.6 


0.4 


0.6 


0.9 


1.0 


1.1 


1.1 


1.1 


1.0 


500 


1.1 


1.0 


O.S 


0.4 


0.2 


0.5 


1.0 


1.3 


1.3 


1.2 


1.1 


600 


1.2 


1.1 


0.9 


0.6 


0.2 


0.2 


0.5 


1.1 


1.5 


1.5 


1.2 


700 


1.4 


1.1 


1.0 


0.8 


0.4 


0.1 


0.3 


0.8 


1.4 


1.7 


1.4 


800 


1.6 


1.3 


1.0 


0.8 


0.6 


0.4 


0.1 


0.3 


1.0 


1.6 


1.6 


900 


1.5 


1.4 


1.1 


0.9 


0.7 


0.6 


0.3 


0.2 


0.6 


1.2 


1.5 


1000 


1.2 


1.5 


1.4 


1.0 


0.7 


0.6 


0.5 


0.5 


0.4 


0.8 


1.2 



Constant, l."0 



TABLE XXXIV. 

Variable Part of Sun's Aberration. 
Argument, Sun's Mean Anomaly. 





O 


Is 


IIS 


Ills 


TVs 


Vs 




o 


// 


,, 


// 


r, 


// 


„ 


o 





0.0 


0.0 


0.1 


0.3 


0.5 


0.6 


30 


3 


0.0 


0.0 


0.2 


0.3 


0.5 


0.6 


27 


6 


0.0 


0.0 


0.2 


0.3 


0.5 


0.6 


24 


9 


0.0 


0.0 


0.2 


0.3 


0.5 


0.6 


21 


12 


0.0 


0.1 


0.2 


0.4 


5 


0.6 


18 


15 


0.0 


0.1 


0.2 


0.4 


0.5 


0.6 


15 


18 


0.0 


0.1 


0.2 


0.4 


0.5 


0.6 


12 


21 


0.0 


0.1 


0.3 


0.4 


0.6 


0.6 


9 


24 


0.0 


0.1 


0.3 


0.4 


0.6 


0.6 


6 


27 


0.0 


0.1 


0.3 


0.4 


0.6 


0.6 


3 


30 


0.0 


0.1 


0.3 


0.5 


0.6 


0.6 







Xls 


X* 


IXs 


VIIIs 


VIIs 


Vis 





Constant, 0."3 



TABLE XXXV. 

Moon's Epochs. 



Years. 



1830 

1831 

1832 B 

1833 

1834 

1835 

1836 B 
1837 
1838 
1839 
1840 B 

1841 
1842 
1843 
1844 B 
1845 

1846 
1847 
1848 B 
1849 
1850 

1851 
1852 B 
1853 
1854 
1855 

1856 B 
1857 
1858 
1859 
1860 B 

1861 
1S62 
1863 
1864 B 
1865 

1866 

1867 
1868 B 
1869 
1870 



00174 
00103 
00032 
00235 
00164 
00093 



4541 
1749 
8957 
6816 
4024 
1232 



4 5 



10 11 I 12 



4461 4638,9885 0635 5979 9921 7623 219 
4127 9331 J2357.6432 7040 '2378 6487 825 
3793 ( 4125[4S29 2229 S100:4S355351 ! 432 
4499 9156 7636 8399 9219 7683 4239:108 
4164 3900 0107 4196 0279 0140 3103715 
3830 8644 257919993 1340. 2598 1967,321 



00022 ,S441 3496 
00224 629914202 
00153 i350Sj3S63 
00032 ; 0716 3534 



00213 
00142 
00071 
00000 
00203 

00132 
00061 
99990 
00192 
00121 



3338 5051 5791 2400 
8419'7858'l960 351S 
3163 0329 7757 4579 



7907^2801 



00011.7925 3199,2651 5273 



5733 3905 
2991J3571 
:0200|3237 
I740S2903 
1-5260^609 

2475i3275 
9633 2941 
I6S92'2606 
4750 3312 
i 1958 2973 



00050 9167 2644 
99979 6375 2310 
00181 4233 3016 
00110 1442 2681 
00039 3650 2347 

9996S 15859 2013 
00171 13717 2719 
0010010925 2335 
00029 JS134 2051 
9995S|5342|l716 

00160 j3200l2423 

000S9 10109 2088 



00018 17617 
9994714826 
00149 12684 



00078 
00007 
99936 
00138 
00067 



1754 
1420 
2126 

1792 
1457 



9893 
7101 
4309 1123 
2168 1829 
9376 1495 



76S2 S080 
2425 0551 
7169 3023 
1913 5495 
6944 8302 

16880773 
6432 3245 
11765717 
6207 8524 
095l|0995 

5695 3467 
0439 5939 
5469 '8746 
0213J1217 

4957.3689 

9701!6160 
473218968 
9476 1439 
42203911 
S9646383 



3555 5639 
9352 6700 



5522 
1319 
7116 
2914 
90S3 2118i8343 



7818 
8879 
9939 
1000 



226.45S 
587 177 



948 

340 
701 
061 



5055 0831 J 928 422 845 
7903 9719, ! 605 814 635 
0360S583;21l'l75 354 
818 7447:818 536 074 
5275 6310 424 896 793 



897 
6S7 
406 

125 



S123 
0580 
3038J 

54951 



5199 101 288 
4062 707 649 
2926| 314 010 
1790(920 371 
0678:597 763 



4SS0 
0678 
6475 
2644 
8442 

4239 
0036 
6206 
2003 
7801 

3598 
9767 
5565 
1362 
7159 



3179 0800 9542 203 
4239 3257 8406|810 
5300(5715 
6418J8563 
747911020 



123 

484 



18539 3477 ' 
9600(5935 
0718'S732 
1 1 778 ! 1240 
2339 3697 

3899'6155 

5013 9002 
,6078 1460 
7139 3917 
!8 199 6374 



7270 416 S45 
6158093 237 
5022)700 597 



3835 
2749 
1637 
0501 



306 

913 

589 

196 
9365 ! 802 

I 
8229 409 793 
7117 086 185 



95S 
319 
711 
072 

432 



5981 
4845 
3709 



3329 
9126 



3995,9190 
8739,1661 
3483J4133 4923 
8227,6605 0721 
325794126890 

8001|1883|2637 
2745 '4355 8485 
7489J68274282 
2520,9634 0452 
726421056249 



9317 9222 2597 581JG59 
0378 1679 1461 188 020 
1438 4137 0324 7951381 
2499 6594 9188 401 742 
3617, 9442 8076 078 134 



692 546 
2991907 

905 267 
,1 



4678 1899 6940 
5738 4357 5804 
6799 6814 4668 
7917 9662 3556 
S97S 2119 2420 



685': 494 
291; 855 
898216 

574 608 
181 968 



583 
302 
022 
741 
531 

250 
970 

689 
479 
199 

918 
637 
427 
147 
866 

586 
375 
095 
814 
534 



468 

940 j 

413 

920 

393 

866 

339 
846 
319 
792 
265 

772 
245 
718 
191 
698 

171 
644 
117 
624 
097 

570 

043 
550 
023 
496 

969 

476 
949 
422 
895 



323 402 



043 
762 
482 
272 

931 
711 
43 1 
220 

"40 



875 
348 
821 
328 

801 

274 
747 
254 
727 



TABLE XXXV. 

Moorfs Epochs. 



40 



Years. 



14 



1830 

1831 

1832 B 

1833 

1834 

1835 

1836 B 
1837 
1838 
1839 
1840 B 

1841 
1842 
1843 
1844 B 
1845 

1846 
1847 
1848 B 
1849 
1850 

1851 
1852 B 
1853 
1854 
1855 

1856 B 
1857 
1858 
1859 
1860 B 

1861 
1862 
1863 
1864 B 
1865 

1866 
1867 
1868 B 
1869 
1870 



921 
115 
309 
602 
796 
989 

183 
476 
670 
864 
058 

351 

544 
738 
932 
225 

419 
613 
806 
099 
293 

487 
681 
974 
168 
361 

555 

848 
042 
236 
430 

723 
916 
110 
304 
597 

791 
985 
178 
471 
665 



15 



392 
532 
673 
844 

984 
124 



16 



230 
5S9 
949 
345 
704 
063 



265 423 
436 819 
576 178 



716 

857 

028 
168 
308 
449 
620 

760 
901 
041 
212 
352 

493 
633 

804 
944 
085 

225 
396 
537 
677 

817 

988 
129 
269 
409 
580 

721 
861 
001 
172 

313 



537 

897 



17 



588 
940 
293 
688 
040 
393 

745 
140 

492 
845 
197 

293 592 
652,944 
012297 
371 649 
767j044 

126j396 
4861749 
845:101 
241496 
600 848 

960 201 
319 553 
715 948 
074300 
434 653 

793 005 

189 400 
548 752 
908!l05 
267457 

663 852 
022204 
382'557 
741 909 
137304 



18 



462 
937 



412-070 



913 

3S8 
S63 

338 
840 
315 
790 

265 

766 
241 
716 
191 
692 

167 
643 
118 
619 
094 

569 
044 
545 
020 
495 

970 

471 
947 

422 
897 

398 
873 
348 
823 
324 



496,657 799 
856 009 274 



215 362 
611(756 
970 109 



749 
251 
726 



19 

523 

296 



845 
619 
392 

166 
942 

715 
489 

262 

038 
811 
585 
358 
134 

907 
681 
454 
230 
003 

777 
550 
326 
099 
873 

646 
422 
195 
969 
742 

518 
291 
065 
838 
614 

387 
161 
934 
710 



20 

536 
703 

870 
037 
203 
370 

537 
704 
870 
037 

204 

371 
537 
704 
871 
038 

204 
371 

538 
705 

871 

038 
205 
372 
539 

705 

872 
039 
206 
372 
539 

706 
873 
039 
206 
373 

540 
707 
873 
040 
207 



21 22 23 



60 44 
70 41 
81 38 
9245 
0342 
1338 

24 35 
35 '42 
4638 
56,35 
67:32 

78 

89 35 
99J32 
10 29 
21 36 



24 25 



27 05 

24 76 
2046 

1717 

2488 
20|60 
17.29 
14 00 
21 71 

17,42 
1412 
1183 
18 54 
1526 



26 27 28 29 30 31 



9253 
40 03 

87 51 

34 01 

85|58 
32 07 



5164 

79J27 
89 



16 08 
65 33 
15'57 
71'86J86 
20.10 10 

70 35 35 
19 59.60 

76 ' 88 88 
2* ! 12|l2 



01 
93 
84 
52176 
1467 



23 61 
8090 
29 15 
79 39 



28 64 64 



84 92 
34 17 
82 41 
32 66 
89 95 



38 19 19 
87 44 44 
37 6869 
93 97 97 
43 21 '21 



91| 15 



G 



50 



TABLE XXXV. 

Moon's Epochs. 



Years. 


Evection. 


Anomaly. 


Variation. 


Longitude. 




8 


o 


f // 


s 


o 


' 


rr 


s 


o 


' 


" 


s 


o 


' 


" 


1830 


5 


17 


4 12 


11 


24 


31 


4.5 


2 


13 


2 


39 


11 


22 


55 


37.7 


1831 


11 


7 


35 41 


2 


23 


14 


24.6 


. 6 


22 


40 


4 


4 


2 


18 


42.8 


1832 B 


4 


38 


7 11 


5 


21 


57 


44.4 


11 


2 


17 


28 


8 


11 


41 


48.0 


1833 


10 


29 


57 40 


9 


3 


44 


58.5 


3 


24 


6 


21 


1 


4 


15 


28.4 


1834 


4 


20 


29 11 





2 


28 


18.5 


8 


3 


43 


45 


5 


13 


38 


33.6 


1835 


10 


ii 


40 


3 


1 


11 


38.6 





13 


21 


10 


9 


23 


1 


38.8 


1836 B 


4 


i 


32 9 


5 


29 


54 


58.7 


4 


22 


58 


34 


2 


2 


24 


44.0 


1837 


10 


3 


22 39 


9 


11 


42 


12.8 


9 


14 


47 


27 


6 


24 


58 


24.5 


1838 


3 


23 


54 9 





10 


25 


32.9 


1 


24 


24 


51 


11 


4 


21 


29.8 


1839 


9 


14 


25 38 


3 


9 


8 


53.1 


6 


4 


2 


16 


3 


13 


44 


35.0 


1840 B 


3 


4 


57 8 


6 


7 


52 


13.2 


10 


13 


39 


42 


7 


23 


7 


40.4 


1841 


9 


6 


47 37 


9 


19 


39 


27.5 


3 


5 


28 


33 





15 


41 


20.9 


1842 


2 


27 


19 7 





18 


22 


47.6 


7 


15 


5 


58 


4 


25 


4 


26.2 


1843 


8 


17 


50 37 


3 


17 


6 


7.9 


11 


24 


43 


23 


9 


4 


27 


31.6 


1844 B 


2 


S 


22 7 


6 


15 


49 


28.1 


4 


4 


20 


48 


1 


13 


50 


37.0 


1845 


8 


10 


12 36 


9 


27 


36 


42.5 


8 


26 


9 


40 


6 


6 


24 


17.5 


1846 


2 





44 6 





26 


20 


2.8 


1 


5 


47 


5 


10 


15 


47 


23.0 


1847 


7 


21 


15 35 


3 


25 


3 


23.2 


5 


15 


24 


30 


2 


25 


10 


28.3 


1848 B 


1 


11 


47 5 


6 


23 


46 


43.5 


9 


25 


1 


55 


7 


4 


33 


33.7 


1849 


7 


13 


37 35 


10 


5 


33 


57.9 


2 


16 


50 


47 


11 


27 


7 


14.5 


1850 


1 


4 


9 4 


1 


4 


17 


18.3 


6 


26 


28 


12 


4 


6 


30 


19.9 


1851 


6 


24 


40 35 


4 


3 





38.6 


11 


6 


5 


37 


8 


15 


53 


25.4 


1852 B 





15 


12 5 


7 


1 


43 


59.2 


3 


15 


43 


3 





25 


16 


31.0 


1853 


6 


17 


2 34 


10 


13 


31 


13.7 


8 


7 


31 


54 


5 


17 


50 


11.6 


1854 





7 


34 4 


1 


12 


14 


34.1 





17 


9 


20 


9 


27 


13 


17.2 


1855 


5 


28 


5 33 


4 


10 


57 


54.7 


4 


26 


46 


44 


2 


6 


36 


22.7 


1856 B 


11 


18 


37 3 


7 


9 


41 


15.2 


9 


6 


24 


10 


6 


15 


59 


28.2 


1857 


5 


20 


27 33 


10 


21 


28 


29.8 


1 


28 


13 


2 


11 


8 


33 


9.1 


1S58 


11 


10 


59 2 


1 


20 


11 


50.3 


6 


7 


50 


27 


3 


17 


56 


14.6 


1859 


5 


1 


30 33 


4 


18 


55 


10.9 


10 


17 


27 


53 


7 


27 


19 


20.1 


1860 B 


10 


22 


2 3 


7 


17 


38 


31.4 


2 


27 


5 


18 





6 


42 


25.8 


1861 


4 


23 


52 32 


10 


29 


25 


46.1 


7 


IS 


54 


10 


4 


29 


16 


6.6 


1862 


10 


14 


24 2 


1 


28 


9 


6.6 


11 


23 


31 


35 


9 


8 


39 


12.2 


1863 


4 


4 


55 32 


4 


26 


52 


27.3 


4 


8 


9 


1 


1 


18 


2 


17.9 


1864 B 


9 


25 


27 2 


7 


25 


35 


48.0 


8 


17 


46 


25 


5 


27 


25 


23.5 


1865 


3 


27 


17 31 


11 


7 


23 


2.7 


I 


9 


35 


18 


10 


19 


59 


4.3 


1866 


9 


17 


49 2 


2 


6 


6 


23.3 


5 


19 


12 


43 


2 


29 


22 


10.1 


1867 


3 


8 


20 31 


5 


4 


49 


44.0 


9 


28 


50 


9 


7 


S 


45 


1 5 . 7 


1868 B 


8 


28 


52 2 


8 


3 


33 


4.7 


2 


8 


27 


34 


11 


IS 


8 


21.4 


1869 


3 





42 33 


11 


15 


20 


19.6 


7 





16 


26 


4 


10 


42 


23 


1870 


8 


21 


14 2 


2 


14 


3 


40.3 


11 


9 


53 


51 


8 


20 


5 


8.0 1 



TABLE XXXV. 



51 



Moon's Epochs. 



Years. 


Supp. of Node. 


II 


V 


VI 


VII 


VIII 


IX 


X 


XI 


XII 


1830 


s o f » 
6 7 7 11.0 


* ° " 
10 24 46 


498 


502 


900 


904 


427 


062 


025 


433 


1831 


6 26 26 53.3 


2 15 18 


912 


914 


208 


210 


506 


001 


211 


710 


1832 B 


7 15 46 35.5 


6 5 50 


326 


327 


516 


516 


586 


940 


397 


986 


1833 


8 5 9 28.4 


10 7 31 


774 


779 


852 


856 


702 


885 


624 


297 


1834 


8 24 29 10.7 


1 28 3 


187 


191 


159 


163 


782 


825 


810 


573 


1835 


9 13 48 53.0 


5 18 35 


601 


603 


467 


469 


861 


764 


996 


850 


1836 B 


10 3 8 35.2 


9 9 8 


015 


016 


775 


775 


941 


703 


182 


127 


1837 


10 22 31 28.1 


1 10 49 


463 


468 


111 


116 


057 


648 


409 


437 


183S 


11 11 51 10.4 


5 1 21 


876 


880 


419 


423 


137 


588 


595 


714 


1839 


1 10 52.6 


8 21 53 


290 


292 


726 


729 


217 


527 


781 


991 


1840 B 


20 30 34.9 


12 25 


704 


705 


034 


035 


296 


466 


967 


268 


1841 


1 9 53 27.7 


4 14 6 


152 


157 


370 


375 


412 


411 


194 


578 


1842 


1 29 13 10.0 


8 4 38 


566 


569 


678 


682 


492 


350 


380 


855 


1843 


2 18 32 52.2 


11 25 10 


980 


980 


986 


988 


572 


290 


566 


131 


1844 B 


3 7 52 34.5 


3 15 42 


393 


394 


293 


294 


651 


229 


752 


408 


1845 


3 27 15 27.4 


7 17 23 


840 


846 


629 


634 


767 


174 


979 


718 


1846 


4 16 35 9.6 


11 7 55 


254 


258 


937 


941 


847 


113 


165 


995 


1847 


5 5 54 51.8 


2 28 27 


668 


670 


245 


247 


927 


053 


351 


272 


1848 B 


5 25 14 34.1 


6 18 59 


082 


083 


553 


553 


006 


992 


537 


549 


1849 


6 14 37 27.0 


10 20 40 


531 


535 


889 


893 


122 


937 


764 


859 


1850 


7 3 57 9.2 


2 11 12 


944 


947 


196 


200 


202 


876 


950 


136 


1851 


7 23 16 51.5 


6 144 


358 


359 


504 


506 


282 


816 


136 


413 


1852 B 


8 12 36 33.6 


9 22 17 


772 


772 


812 


812 


362 


755 


322 


689 


1853 


9 1 59 26.5 


123 58 


220 


223 


148 


152 


477 


700 


549 


000 


1854 


9 21 19 8.8 


5 14 30 


634 


636 


456 


459 


557 


639 


735 


276 


1855 


10 10 38 51.1 


9 5 2 


047 


048 


763 


765 


637 


579 


921 


553 


1856 B 


10 29 58 33.3 


25 34 


461 


461 


71 


071 


717 


518 


107 


830 


1857 


11 19 21 26.2 


4 27 15 


909 


912 


407 


411 


832 


463 


334 


140 


1858 


8 41 8.4 


8 17 47 


323 


325 


715 


718 


912 


402 


520 


417 


1859 


28 50.7 


8 19 


736 


737 


023 


024 


992 


342 


706 


694 


1860 B 


1 17 20 32.9 


3 28 51 


150 


150 


330 


330 


072 


281 


892 


971 


1861 


2 6 43 25.8 


8 32 


598 


601 


666 


670 


187 


226 


119 


281 


1862 


2 26 3 8.0 


11 21 4 


012 


014 


974 


977 


267 


165 


305 


558 


1863 


3 15 22 50.1 


- 3 11 36 


426 


426 


282 


283 


347 


105 


491 


834 


1864 B 


4 4 42 32.3 


7 2 8 


839 


839 


590 


589 


427 


044 


677 


111 


1865 


4 24 5 25.2 


11 3 49 


287 


291 


926 


929 


542 


989 


904 


422 


1866 


5 13 25 7.3 


2 24 21 


701 


703 


233 


236 


622 


928 


090 


698 


1867 


6 2 44 49.5 


6 14 53 


115 


115 


541 


542 


702 


868 


276 


975 


1868 B 


6 22 4 31.7 


10 5 26 


529 


528 


849 


848 


782 


807 


462 


252 


1869 


7 11 27 24.6 


2 7 7 


977 


980 


185 


188 


897 


752 


689 


562 


! 1870 


8 47 6.7 


5 27 39 


390 


392 


493 


495 


977 


691 


875 


839 



52 



TABLE XXXVI. 

Moorfs Motions for Months. 



f Months. 



January 
February- 
March 



April 
May 
June 

July 
Aug. 

Sept. 

Oct. 
Nov. 
Dec. 



Com. 
Bis. 

( Com. 
{Bis. 
( Com. 
{Bis. 
( Com. 
1 Bis. 

c Com. 
{Bis. 
( Com. 
{Bis. 
( Com. 
{ Bis. 

< Com. 
f Bis. 
( Com. 
{ Bis. 
j Com. 
{Bis. 



00000 
08487 
16153 
16427 

24640 
24914 
32853 
33127 
41340 
41614 

49554 
49828 
58041 
58315 
66528 
66802 

74741 
75015 
83228 
83502 
91442 
91716 



2 3 



0000 
0146 
8343 
8993 

8490 
9140 
7986 
8636 
8133 
8783 

7629 
8279 
7776 

8426 
7922 
8572 

7419 
8069 
7565 
8215 
7062 
7712 



0000 
2246 
1371 
2411 

3616 
4657 

4822 



0000 

8896 
6931 
7218 

5827 
6114 
4436 



706713332 

810713619 

! 
8273 1942 
931312228 



0518 
1558 
2764 
3804 

3969 
5009 
6215 
7255 
7420 
8460 



0838 
1125 
9734 
0021 

8343 
8630 
7239 
7526 
5848 
6135 



0000 
0402 
9797 
0132 

0199 
0534 
0265 
0600 
0666 
1002 

0732 
1068 



0000 
1533 
1951 
2323 

3484 
3856 
4646 
5018 
6179 
6551 

7341 
7713 



8 9 10 11 12 13 



0000 0000 
1789 2099 
3404 3027 
3462 3418 



1134|8874 
1470 9246 
1536 0408 
1871 0780 



1602 
1938 
2004 
2339 
2070 
2405 



1569 
1941 
3102 
3475 
4264 
4636 



5193 
5251 
6924 
6982 
8713 
8771 

0444 
0502 
2233 
2290 
4021 
4079 

5752 
5810 
7541 
7599 
9272 
9330 



5126 
5517 
6835 
7226 
8934 
9325 



0000 
0753 
1433 

1457 

2186 
2210 
2914 
2938 
3667 
3691 



1034 
2742 
3133 
4842 
5232 



0643;4396 
4420 
5148 
5173 
5901 
5925 



65506630 
6941 6654 
8649 7382 
904017407 
03588111 
0749 8135 



000 

175 
139 
209 

314 
384 
419 
489 
593 
663 

698 
768 
873 
943 
048 
118 

152 
222 
327 
397 
432 
502 



000 
965 

836 

868 

801 
832 
735 
766 
700 
731 

634 
665 
599 
630 
563 
595 

497 
528 
462 
493 
396 
427 



000 

184 
157 

228 

342 

412 
456 
526 
640 
710 

754 
824 
938 
009 
123 
193 

237 
307 
421 
492 
535 
606 



000 

059 
016 
050 

076 
110 
101 
135 
160 
194 

185 
219 
245 
279 
304 
338 

329 
363 

388 
423 
414 
448 









TABLE XXXVI. 








Moon! 


$ Motions for 


Months. 




Months. 


Evection. 


Anomaly. 


Variation. 


Longitude. 


January 


3 ° ' " 



g O ' " 

0.0 


s ° ' " 



0.0 


February 


11 20 48 42 


1 15 53.1 


17 54 48 


1 18 28 5.8 


March 


( Com. 
{Bis. 


10 7 40 26 


1 20 50 4.2 


11 29 15 15 


1 27 24 26.6 


10 18 59 26 


2 3 53 58.2 


11 26 42 


2 10 35 1.6 


April 


5 Com. 
1 Bis. 


9 28 29 8 


3 5 50 57.3 


17 10 3 


3 15 52 32.5 


10 9 48 8 


3 18 54 51.2 


29 21 29 


3 29 3 7.5 


May 


( Com. 
{Bis. 


9 7 58 51 


4 7 47 56.4 


22 53 24 


4 21 10 3.3 


9 19 17 50 


4 20 51 50.3 


1 5 4 50 


5 4 20 38.3 


June 


( Com. 
{Bis. 


8 28 47 33 


5 22 48 49.4 


1 10 48 11 


6 9 38 9.1 


9 10 6 33 


6 5 52 43.4 


1 22 59 38 


6 22 48 44.1 


July 


( Com. 
{ Bis. 


8 8 17 16 


6 24 45 48.5 


1 16 31 32 


7 14 55 39.9 


8 19 36 15 


7 7 49 42.5 


1 28 42 59 


7 28 6 15.0 


Aug. 


< Com. 
{Bis. 


7 29 5 59 


8 9 46 41.6 


2 4 26 20 


9 3 23 45.8 


8 10 24 58 


8 22 50 35.5 


2 16 37 47 


9 16 34 20.8 


Sept. 


( Com. 
{Bis. 


7 19 54 41 


9 24 47 34.6 


2 22 21 7 


10 21 51 51.6 


8 1 13 40 


10 7 51 28.6 


3 4 32 34 


11 5 2 26.7 


Oct. 


( Com. 
{Bis. 


6 29 24 24 


10 26 44 33.7 


2 28 4 28 


11 27 9 22.4 


7 10 43 23 


11 9 48 27.7 


3 10 15 55 


10 19 57.5 


Nov. 


( Com. 
{Bis. 


6 20 13 6 


11 45 26.8 


3 15 59 16 


1 15 37 28.3 


7 1 32 5 


24 49 20.7 


3 28 10 43 


1 28 48 3.3 


Dec. 


( Com. 
{Bis. 


5 29 42 49 


1 13 42 25.9 


3 21 42 37 


2 20 54 59.1 


6 11 1 48 


1 26 46 19.8 


4 3 54 4 


3 4 5 34.1 



TABLE XXXVI. 

Moon's Motions for Months. 



53 



Months. 



January 
February 

March 



April 
May 

June 

July 
Aug. 
Sept. 

Oct. 
Nov. 
Dec. 



Com. 
. Bis. 

( Com. 
1 Bis. 
( Com. 
t Bis. 
( Com. 
tBis. 

( Com. 
{Bis. 
( Com. 
tBis. 
( Com. 
1 Bis. 

( Com. 
i Bis. 
j Com. 
\ Bis. 
( Com. 
>Bis. 



14 15 16 I 17 , 18 I 19 I 20|21|22|23 24i25 26;27|28]29 30 31 



000 
074 
851 
950 

925 



000 
946 
SOI 
831 

747 



000 
135 
159 
196 

294J786 



024, 77S 
899 663 
999 693 4291089 



l;331 828 
''392 047 



000 000 000 000 oojoo 
304i805 066 014! 24.26 
482 532,125 027; 45;50 
524 55S]l27 027 

191041 
193 042 



973 
073 

94S 
047 
022 
121 
096 
195 

071 
170 
145 
244 
120 
219 



609 
639 

525 
555 
471 
501 
417 
447 

333 
363 
279 
309 



527 351 
563 393 



625 
661 
759 
796 
894 
931 

992 
029 

127 
163 



194 225 
225'261 



46 51 



68 77 

69 77 



336 

362 

115*254 055; 91 02 

141 

920 
946 



613 699 
655 725 
917 503 
959 529 
221 308 
263.334 

483,087 
525 1 13 
7S7 ( 892 
S29 918 
049 670 
091 696 



J256 055 92 03 
320 069] 15 28 
322 069 15129 

384 083,37 54 

386083 3855 



449 097 
451,097 
515- Ill 
517 111 



57S 
581 

644 
646 
70S 
710 



125 
126 
139 
140 
153 
153 



85 07 

85 08 



00 00 

14.82 
98 57 
08 59 

1239 
22|42 
15 19 
26:22 
29:01 
40 04 

33181 
43184 
47:64 
57 66 
61 46 
71 49 



oo'oooo 

14 17 29 

18 12 46 

47|21 19J51 

70 32 29,76 
74j36 36 80 
94*43 38 01 



98 47 

21,57 
2561 

45,68 
49 72 
72 82 
77,86 
00 97 
0401 

26|23 08 
28|28'll 
0851,22 

11J55 26 
83188 74 33 
9319079 37 



45 05 
55 31 
62 35 

I 
64 56 
71 60 
81 85 
88 90 
97 15 
04 19 

07 40 
14 44 
23 70 
30 74 
33 95 
40 99 



10 15 

15;23 
16.23 

21130 
2l]30 
26 38 
2638 

3145 

31 46 
36 53 
36 53 

42 61 
42 61 

47 68 
47 69 
52 76 
52 76 

57 84 
57 84 



TABLE XXXVI. 

Moorts Motions for Months. 



Months. 



January 
February 

March 



April 
May 
June 

July 
Aug. 
Sept. 

Oct. 
Nov. 
Dec. 



Com. 
Bis. 

Com. 

Bis. 

Com. 

Bis. 

Com. 

Bis. 

Com. 

Bis. 

Com, 

Bis. 

Com. 

Bis. 

Com. 

Bis. 

Com, 

Bs. 

Com 

Bis. 



Supp. of Node.' 



II 



0.0 

1 38 29.7 
3 7 27,5 

3 10 38.2 

4 45 57.3 
4 49 7.9 
6 21 16.4 

6 24 27.0 

7 59 46.1 

8 2 56.7 



V VI VII VIII IX X II XI 1XII 



9 35 5.2 

"9 38 15.9 

11 13 35.0 

11 16 45.6 

12 52 4.7 

12 55 15.4 

14 27 23.8 

14 30 34.4 

16 5 53.5 

16 9 4.2 

17 41 12.6 

17 44 23.3 



000 000 
15 43 1 054 224 
27 59 I 007, 330 

9 8 \ 041 369 

13 42 I 061 ! 554 
24 51 J 095] 593 

18 15 08l]738 
8 29 25 ! 115 778 
8 3 58 136 962 
8 15 8 170 002 

7 8 32 156 147 
7 19 41 190 186 

6 24 15 210 371 

7 5 24 I 244j 411 
6 9 58 1 265 595 
6 21 7 299 635 



5 14 32 
5 25 41 
5 15 
5 11 24 
4 4 49 
4 15 58 



285 
319 
339 
373 
359 
393 



780 
819 
004 
043 

188 
228 



000 
875 
666 
694 

542 
570 
3S9 
417 
264 
293 

112 
140 
987 
015 
862 
891 

710 
738 
585 
613 
432 
461 



000 
045 
9S9 
023 

034 
068 
046 
080 
091 
124 

103 
136 
147 
182 
193 
227 

204 
238 
250 
283 
261 
295 



000 
111 

114 



000 
165 
313 



150 319 



225 
261 
300 
336 
411 
447 



478 
484 
638 
643 
802 
808 



745 
787 
993 
1034 
282 
324 

531 
572 
820 
862 
110 
152 

783 451 358 
819 456 400 
894 i 615 |648 
690 
896 
938 



486 962 
522 I 967 
597 126 



633 

708 
744 



132 
291 
296 



000 j 000 
290 i 043 
455 j 984 
496 , 018 



930 j 621 
969 775 
005 I 780 



027 
061 
036 
070 
079 
113 

088 
122 
131 
164 
173 
207 

182 
216 
225 
259 
234 



54 



TAELE XXXVIL 



MoorCs Motions for Days, 



D. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


10 


11 


12 


13 


1 


00000 


0000 


0000 


0000 I 0000 


0000 


0000 0000 


0000 


000 


000 


000 


006 i 


2 


00274 


0650 


1040 


0287 [ 0336 


0372 


0058 


0390 


0024 


070 


031 


070 


034; 


3 


00548 


1300 


2080 


0574 ! 0671 


0744 


0115 


0781 


0049 


140 


062 


141 


068 


4 


00S21 


1950 


3121 


0861 


1007 


1116 


0173 


1171 


0073 


210 


093 


211 


103 


5 


01095 


2600 


4161 


1148 


1342 


1488 


0231 


1561 


0097 


281 


125 


282 


137 


6 


01369 


3249 


5203 


1435 1678 1860 0289 \ 1952 


0121 


351 


156 


352 


171 


7 


01643 


3S99 


6241 


1722 2013 


2232 


0346 2342 


0146 


421 


187 


423 


205 


8 


01916 


4549 


7281 


2009 


2349 


2604 


0404 


2732 


01701491 


218 


493 


239 


9 


02190 


5199 


8321 


2296 


2684 


2976 


0462 


3122 


0194 ! 561 


249 


564 


273 


10 


02464 


5849 


9362 


25S3 


3020 


3345 


0519 


3513 


0219 t 631 


280 


634 


308 


11 


02738 


6499 


0402 


2870 


3355 


3720 


0577 


3903 


0243 j 702 


311 


705 


342 


12 


03012 


7149 


1442 


3157 3691 


4093 


0635 


4293 


0267 772 


342 


775 


376 


13 


03285 


7799 


2482 


3444 4026 


4465 


0692 


4684 


0291i 842 


374 


845 


410 


14 


03559 


8449 


3522 


3731 J 4362 


4837 


0750 


5074 0316 '912 


405 


916 


444 


15 


03833 


9098 


4563 


4018 


46 9S 


5209 


0S08 


5464 0340 982 


436 


986 


478 


10 


04107 


9748 


5603 


4305 


5033 


5581 


0866 


5854 0364 ' 052 


467 


057 


513 


17 


04380 


0398 


6643 


4592 


5369 


5953 


0923 


6245,0389 122 


498 


127 


547 


18 


04654 


1048 


7683 


4878 


5704 


6325 


0981 


6635 0413 


193 


529 


198 


581 


19 


0492S 


1698 


8723 


5165 


6040 


6697 


1039 


7025 0437 


263 


560 


268 


315 


20 


05202 


2348 


9763 


5452 


6375 


7069 


1096 


7416 0461 


333 


591 


339 


649 


21 


05476 


2998 


0804 


5739 


6711 


7441 


1154 


7806 0486 


403 


623 


409 


683 


22 


05749 


3648 


1844 


6026 


7046 


7813 


1212 


8196; 0510 


473 


654 


4S0 


718 


23 


06023 


4298 


2SS4 


6313 


7382 


8185 


1269 8586 | 0534 


543 


685 


550 


752 


24 


06297 


4947 


3924 


6600 


7717 


8557 


1327 


S977 10559 


614 


716 


621 


786 


25 


06571 


5597 


4964 


6887 8053 


8929 


1385 


9367 0583 


684 


747 


691 


820 


26 


06844 


6247 6005 


7174' 83S9 


9301 1443 


9757 0607 


754 


778 


762 


854 


27 


07118 


6897 ' 7045 


7461 8724 


9673 1500 


0148 0631 


824 


809 


832 888 


28 


07392 


7547 S0S5 7748 9060 


0045 j 1558 


0538 0656 


894 


840 


903 923 


' 29 


07666 


S I 97 9125! 8035,9395 


0417] 1616 | 0928 j 0680 


964 


872 


973 957 1 


30 


07940 


8847 0165 1 8322 1 9731 0789 ] 1673 j 1?I9 j 0704 


034 


903 


043 991 1 


. 31 


0S213 


9497 ! 1205 ( 8609 00661 116111731 [170910729 


105 


934 


114|025 



TABLE XXXVII 



55 



Moon's Motion for Days. 



D 


14 


15 


16 


17 


18 


19 


20 


21 


22 


23 


24 


25 


26 


27 


28 


29 


30 


31 





___ 


: 













— 


— 


— 


— 


— 


— 


— 


— 


— 


— 


— 





1 


000 


000 


000 


000 


000 


000 


000 


00 


00 


00 


00 


00 


00 


00 


00 


00 


00 


00 


2 


099 


031 


037 


042 


026 


002 


000 


01 


01 


10 


03 


04 


04 


or 


04 


03 


00 


00 


3 


198 


061 


073 


084 


052 


004 


001 


02 


02 


20 


05 


08 


07 


14 


08 


06 


00 


00 


4 


297 


092 


110 


126 


078 


006 


00] 


02 


03 


30 


03 


12 


11 


21 


13 


09 


01 


01 


5 


397 


122 


146 


168 


104 


008 


002 


03 


03 


41 


11 


16 


15 


28 


17 


12 


01 


01 


6 


496 


153 


183 


210 


130 


Oil 


002 


04 


04 


51 


13 


21 


18 


35 


21 


15 


01 


01 


7 


595 


183 


220 


252 


156 


013 


003 


05 


05 


61 


16 


25 


22 


42 


25 


18 


01 


01 


8 


694 


214 


256 


294 


182 


015 


003 


05 


03 


71 


19 


29 


26 


49 


29 


22 


01 


02 


9 


793 


244 


293 


336 


208 


017 


004 


06 


07 


81 


21 


33 


30 


56 


33 


25 


01 


02 


10 


892 


275 


329 


379 


234 


019 


004 


07 


03 


91 


24 


37 


33 


63 


38 


28 


02 


02 


11 


992 


305 


366 


421 


260 


021 


005 


08 


09 


01 


27 


41 


37 


70 


42 


31 


02 


02 


12 


091 


336 


403 


463 


286 


023 


005 


08 


09 


11 


29 


45 


41 


77 


46 


34 


02 


03 


13 


190 


366 


439 


505 


312 


025 


005 


09 


10 


22 


32 


49 


44 


84 


50 


37 


02 


03 


14 


289 


397 


476 


547 


337 


028 


006 


10 


11 


32 


34 


53 


43 


91 


54 


40 


02 


03 


15 


388 


427 


512 


589 


363 


030 


006 


11 


12 


42 


37 


58 


52 


98 


58 


43 


02 


03 


16 


487 


458 


549 


631 


389 


032 


007 


11 


13 


52 


40 


62 


55 


05 


63 


46 


03 


04 


17 


587 


488 


586 


673 


415 


034 


007 


12 


14 


32 


42 


66 


59 


12 


67 


49 


03 


04 


18 


686 


519 


622 


715 441 


036 


008 


13 


14 


72 


45 


70 


63 


19 


71 


52 


03 


04 


19 


785 


549 


659 


757 467 


038 


008 


14 


15 


82 


48 


74 


66 


26 


75 


55 


03 


04 


20 


884 


580 


695 


799 493 


040 


009 


14 


16 


92 


50 


78 


70 


33 


79 


59 


03 


05 


21 


983 


611 


732 


841 519 


042 


009 


15 


17 


03 


53 


82 


74 


40 


84 


62 


03 


05 


22 


082 


641 


769 


883 545 


044 


010 


16 


IS 


13 


56 


86 


77 


47 


88 


65 


04 


05 


23 


182 


672 


805 


925 571 


047 


010 


17 


13 


23 


58 


90 


31 


54 


92 


3N 


04 


05 


24 


281 


702 


842 


967 


597 


049 


Oil 


17 


20 


33 


61 


95 


85 


61 


96 


71 


04 


06 


25 


380 


733 


878 


009 


623 


051 


Oil 


18 


20 


43 


64 


99 


89 


68 


00 


74 


04 


06 


26 


479 


763 


915 


052 


649 


053 


Oil 


19 


21 


53 


66 


03 


92 


75 


04 


77 


04 


06 


27 


578 


794 


952 


094 


675 


055 


012 


20 


22 


63 


69 


07 


96 


82 


09 


SO 


04 


06 


28 


677 


824 


988 


136 


701 


057 


012 


20 


23 


73 


72 


11 


00 


89 


13 


S3 


05 


06 


29 


777 


855 


025 


178 


727 


059 


013 1 


21 


24 


84 


74 


15 


03 


96 


17 


86 


05 


07 


30 


876 


885 


061 


220 


753 


061 


013 1 


22 


25 


94 


77 


19 


07 


03 


21 


S9 


05 


07 


31 


975 


916 


098 


262 


779 


064' 014 [ 


23 


26 


04 


8 -J 


23 


11 


10 


25 


92 


05 


07 



56 



TABLE XXXVII. 



MoorCs Motions for Days. 



D. 


Evection. 


Anomaly. 


Variation. 


— 

M. Longitude. 


1 


s ° ' 




s o ' " 

00 


8 ° ' " 




s 9 ' " 
00 


2 


11 18 59 


13 3 54.0 


.0 12 11 27 


13 10 35.0 


3 


22 37 59 


26 7 47.9 


24 22 53 


26 21 10.1 


4 


1 3 56 58 


1 9 11 41.9 


1 6 34 20 


1 9 31 45.1 


5 


1 15 15 58 


1 22 15 35.9 


1 18 45 47 


1 22 42 20.1 


6 


1 26 34 57 


2 5 19 29.8 


2 57 13 


2 5 52 55.1 


7 


2 7 53 57 


2 18 23 23.8 


2 13 8 40 


2 19 3 30.2 


8 


2 19 12 56 


3 1 27 17.8 


2 25 20 7 


3 2 14 5.2 


9 


3 31 55 


3 14 31 11.7 


3 7 31 34 


3 15 24 40.2 


10 


3 11 50 55 


3 27 35 5.7 


3 19 43 


3 28 35 15.2 


11 


3 23 9 54 


4 10 38 59.7 


4 1 54 27 


4 11 45 50.3 


12 


4 4 28 54 


4 23 42 53.7 


4 14 5 54 


4 24 56 25.3 


13 


4 15 47 53 


5 6 46 47.6 


4 26 17 20 


5 8 7 0.3 


14 


4 27 6 53 


5 19 50 41.6 


5 8 28 47 


5 21 17 35.4 


15 


5 8 25 52 


6 2 54 35.6 


5 20 40 14 


6 4 28 10.4 


16 


5 19 44 51 


6 15 58 29.5 


6 2 51 40 


6 17 38 45.4 


17 


6 1 3 51 


6 29 2 23.5 


6 15 3 7 


7 49 20.4 


18 


6 12 22 50 


7 12 6 17.5 


6 27 14 34 


7 13 59 55.5 


19 


6 23 41 50 


7 25 10 11.4 


7 9 26 1 


7 27 10 30.5 


20 


7 5 49 


8 8 14 5.4 


7 21 37 27 


8 10 21 5.5 


21 


7 16 19 49 


8 21 17 59.4 


8 3 48 54 


8 23 31 40.5 


22 


7 27 38 48 


9 4 21 53.4 


8 16 21 


9 6 42 15.6 


23 


8 8 57 47 


9 17 25 47.3 


8 28 11 47 


9 19 52 50.6 


24 


8 20 16 47 


10 29 41.3 


9 10 23 14 


10 3 3 25.6 


25 


9 1 35 46 


10 13 33 35.3 


9 22 34 41 


10 16 14 0.7 


26 


9 12 54 46 


10 26 37 29.2 


10 4 46 7 


10 29 24 35.7 


27 


9 24 13 45 


11 9 41 23.2 


10 16 57 34 


11 12 35 10.7 


28 


10 5 32 45 


11 22 45 17.2 


10 29 9 1 


11 25 45 45.7 


29 


10 16 51 44 


0. 5 49 11.1 


11 11 20 28 


8 56 20.8 


30 


10 28 10 43 


18 53 5.1 


11 23 31 54 


22 6 55.8 


31 I 


11 9 29 43 


1 1 56 59.1 


5 43 21 


1 5 17 30.8 



TABLE. XXXVII. 



57 



Mooris Motions for Days. 



D 


Supp. of Node. 


II 


V 


VI 


VII 


VIII 


IX 


X 


XI 


XII 


1 


s o / '/ i 
0.0 


s ° ' 



000 


000 


000 


000 


000 


000 


000 


000 


2 


3 10.6 


11 9 


034 


039 


028 


034 


036 


005 


042 


034 


3 


6 21.3 


22 18 


068 


079 


056 


067 


072 


Oil 


083 


067 


4 


9 31.9 


1 3 27 


102 


118 


085 


101 


108 


016 


125 


101 


5 


12 42.5 


1 14 37 


136 


158 


113 


135 


143 


021 


166 


135 


6 


15 53.2 


1 25 46 


170 


197 


141 


169 


179 


027 


208 


168 


7 


19 3.8 


2 6 55 


204 


237 


169 


202 


215 


032 


250 


202 


S 


22 14.5 


2 18 4 


238 


276 


198 


236 


251 


037 


291 


235 


9 


25 25.1 


2 29 13 


272 


316 


226 


270 


287 


043 


333 


269 


10 


28 35.7 


3 10 22 


306 


355 


254 


303 


323 


048 


374 


303 


11 


31 46.4 


3 21 31 


340 


395 


282 


337 


358 


053 


416 


336 


12 


34 57.0 


4 2 40 


374 


434 


311 


371 


394 


058 


458 


370 


13 


38 7.6 


4 13 50 


408 


474 


339 


405 


430 


064 


499 


404 


14 


41 18.3 


4 24 59 


442 


513 


367 


438 


466 


069 


541 


437 


15 


44 28.9 


5 6 8 


476 


553 


395 


472 


502 


074 


583 


471 


16 


47 39.5 


5 17 17 


510 


592 


424 


506 


538 


080 


624 


505 


17 


50 50.2 


5 28 26 


544 


632 


452 


539 


573 


085 


666 


538 


IS 


54 0.8 


6 9 35 


578 


671 


480 


573 


609 


090 


707 


572 


19 


57 11.5 


6 20 44 


612 


711 


508 


607 


645 


096 


749 


605 


20 


1 22.1 


7 1 53 


646 


750 


537 


641 


681 


101 


791 


639 


21 


1 3 32.7 


7 13 3 


680 


790 


565 


674 


717 


106 


832 


673 


22 


1 6 43.4 


7 24 12 


714 


829 


593 


708 


753 


112 


874 


706 


23 


1 9 54.0 


8 5 21 


748 


869 


621 


742 


788 


117 


915 


740 


24 


1 13 4.6 


8 16 30 


782 


908 


'650 


775 


824 


122 


957 


774 


25 


1 16 15.3 


8 27 39 


816 


948 


678 


809 


860 


128 


999 


807 


26 


1 19 25.9 


9 8 48 


850 


987 


706 


843 


896 


133 


040 


841 


27 


1 22 36.5 


9 19 57 


884 


027 


734 


877 


932 


138 


082 


875 


28 


1 25 47.2 


10 1 6 


918 


066 


762 


910 


968 


143 


123 


908 


29 


1 28 57.8 


10 12 16 


952 


106 


791 


944 


003 


149 


165 


942 


30 


1 32 8.5 


10 23 25 


986 


145 


819 


978 


039 


154 


207 I 


975 


31 


1 35 19.1 


11 4 34 


020 


1S5 


847 


Oil 


075 


159 


248 1 


009 J 



H 



58 



TABLE XXXVIII. 

Moon's Motions for Hours. 



H. 


1 


2 


3 


1 


11 


27 


43 


2 


23 


54 


87 


3 


34 


81 


130 


4 


46 


108 


173 


5 


57 


135 


217 


6 


68 


162 


260 


7 


80 


190 


303 


8 


91 


217 


347 


9 


103 


244 


390 


10 


114 


271 


433 


11 


125 


298 


477 


12 


137 


325 


520 


13 


148 


352 


563 


14 


160 


379 


607 


15 


171 


406 


650 


16 


182 


433 


693 


17 


194 


460 


737 


18 


205 


487 


780 


19 


217 


515 


823 


20 


228 


542 


867 


21 


239 


569 


' 910 


22 


251 


596 


953 


23 


262 


623 


997 


24 


274 


650 


1040 



8 9 10 11 12 13 



12 
24 
36 

48 

60 

72 
84 
96 

108 
120 
131 
143 

155 
167 
179 
191 

203 
215 
227 
239 

251 
263 
275 

287 



14 
28 
42 
56 

70 

84 

98 

112 

126 
140 
154 
168 

182 
196 
210 

224 

238 
252 
266 
280 

294 
308 
322 
336 



16 
31 

47 
62 

78 

93 

109 

124 

140 
155 

171 
186 

202 
217 
233 

248 

264 
279 
295 
310 

326 
341 
357 
372 



2 

5 

7 

10 

12 
14 
17 
19 

22 
24 
26 
29 

31 
34 
36 
38 

41 
43 
46 
48 

50 
53 
55 

58 



16 
33 

49 
65 

81 
98 
114 
130 
146 
163 
179 
195 

211 

228 
244 
260 

276 
293 
309 
325 

341 
358 
374 
390 



1 
2 
3 
4 

5 

6 

7 

8 

9 
10 
11 
12 

13 

14 
15 
16 

17 
18 
19 
20 

21 
22 
23 



3 

6 
9 

12 
15 
18 
20 
23 

26 
29 
32 
35 

38 
41 
44 

47 

50 
53 

56 

58 

61 
64 

67 



24 | 70 



1 
3 
4 
5 

6 

8 

9 

10 
12 
13 
14 
16 

17 
18 
19 

21 

22 
23 
25 
26 

27 
28 
30 
31 



3 
6 
9 

12 
15 
18 
20 
23 

26 
29 
32 
35 

38 
41 
44 
47 

50 

53 
56 

58 

61 
64 
67 
70 



1 
3 
4 
6 

7 

9 
10 
11 
13 
14 
16 
17 

18 
20 
21 
23 

24 
25 

27 
28 

30 
31 
33 
34 



Hours. 


Evection. 


Anomaly. 


Variation. 


Longitude. 




O r " 


O ' '/ 


O / /' 


O ' " 


1 


28 17 


32 39.7 


30 29 


32 56.5 


2 


56 35 


1 5 19.5 


1 57 


1 5 52.9 


3 


1 24 52 


1 37 59.2 


1 31 26 


1 38 49.4 


4 


1 53 10 


2 10 39.0 


2 1 54 


2 11 45.8 


5 


2 21 27 


2 43 18.7 


2 32 23 


2 44-42.3 


6 


2 49 45 


3 15 58.5 


3 2 52 


3 17 38.8 


7 


3 18 2 


3 48 38.2 


3 33 20 


3 50 35.2 


8 


3 46 20 


4 21 18.0 


4 3 49 


4 23 31.7 


9 


4 14 37 


4 53 57.7 


4 34 17 


4 56 28.1 


10 


4 42 55 


5 26 37.5 


5 4 46 


5 29 24.6 


11 


5 11 12 


5 59 17.2 


5 35 15 


6 2 21.0 


12 


5 39 30 


6 31 57.0 


6 5 43 


6 35 17.5 


13 


6 7 47 


7 4 36.7 


6 36 12 


7 8 14.0 


14 


6 36 5 


7 37 16.5 


7 6 40 


7 41 10.4 


15 


7 4 22 


8 9 56.2 


7 37 9 


8 14 6.9 


16 


7 32 40 - 


8 42 36.0 


8 7 38 


8 47 3.4 


17 


8 57 


9 15 15.7 


8 38 6 


9 19 59.8 


18 


8 29 15 


9 47 55.5 


9 8 35 


9 52 56.3 


19 


8 57 32 


10 20 35.2 


9 39 3 


10 25 52.7 


20 


9 25 50 


10 53 15.0 


10 9 32 


10 58 49.2 


21 


9 54 7 


11 25 54.7 


10 40 1 


11 31 45.6 


22 


10 22 24 


11 58 34.5 


11 10 29 


12 4 42.1 


23 


10 50 42 


12 31 14.2 


11 40 58 


12 37 38.6 


24 


11 18 59 


13 3 54.0 


12 11 27 


| 13 10 35.0 



TABLE. XXXVIII. 



59 



Moon's Motions for Hours. 



3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 
14 
15 
16 

17 
18 
19 
20 

21 
22 
23 
24 



14 


15 


16 


17 


18 | 


19 


20 


21 


22 


23 


24 


4 


1 


2 


2 


1 




















8 


3 


3 


4 


2 














1 





12 


4 


5 


5 


3 














1 





16 


5 


6 


7 


4 














2 





21 


6 


8 


9 


5 














2 


1 


25 


8 


9 


11 


6 














3 


1 


29 


9 


11 


12 


8 


1 











3 


1 


33 


10 


12 


14 


9 


1 











3 


1 


37 


11 


14 


16 


10 


1 











4 


1 


41 


13 


15 


18 


11 


1 











4 


1 


45 


14 


17 


19 


12 


1 











5 


1 


49 


15 


18 


21 


13 


1 











5 


1 


54 


16 


20 


23 


14 


1 











5 


1 


58 


18 


21 


25 


15 


1 











6 


2 


62 


19 


23 


26 


16 


1 











6 


2 


66 


20 


25 


28 


17 


1 





1 


1 


7 


2 


70 


21 


26 


30 


18 


1 









7 


2 


74 


23 


28 


32 


19 


2 









8 


2 


78 


24 


29 


33 


21 


2 









8 


2 


83 


25 


31 


35 


22 


2 









8 


2 


87 


26 


32 


37 


23 


2 









9 


2 


91 


28 


34 


39 


24 


2 


1 






9 


2 


95 


29 


35 


40 


25 


i 2 









10 


3 


99 


! 31 


1 37 


42 


26 


1 2 


1 o 






1 io 


3 





1 
1 
1 

1 

2 
2 
2 

3 
3 

3 

3 

4 
4 
4 

5 

5 
5 
6 
6 

6 
6 

7 

7 



1 
2 
3 
4 

5 

6 
7 
8 

9 

10 
11 
12 

13 
14 
15 
16 

17 
18 

19 
20 

21 
22 
23 

24 



Sup. of Nod. 


II 


V 


VI 


VII 


VIII 


IX 


X 


XI 


XII 


7.9 


O ' 

28 


1 


2 


1 


1 


1 





2 


1 


15.9 


56 


3 


3 


2 


3 


3 





3 


3 


23.8 


1 24 


4 


5 


4 


4 


4 


1 


5 


4 


3J. 8 


1 52 f 


6 


7 


5 


6 


6 


1 


7 


6 


39.7 


2 19 


7 


8 


6 


7 


7 


1 


9 


7 


47.7 


2 47 


9 


10 


7 


9 


9 


1 


10 


9 


55.6 


3 15 


10 


12 


8 


10 


10 


2 


12 


10 


1 3.6 


3 43 


11 


13 


9 


11 


12 


2 


14 


11 


1 11.5 


4 11 


13 


15 


11 


13 


13 


2 


15 


13 


1 19.4 


4 39 


14 


16 


12 


14 


15 


2 


17 


14 


1 27.4 


5 7 


16 


18 


13 


15 


16 


2 


19 


15 


1 35.3 


5 35 


17 


20 


14 


17 


18 


3 


21 


17 


1 43.3 


6 2 


18 


21 


15 


18 


19 


3 


23 


18 


1 51.2 


6 30 


20 


23 


16 


19 


21 


3 


24 


19 


1 59.2 


6 58 


21 


25 


18 


21 


22 


3 


26 


21 


2 7.1 


7 26 


23 


26 


19 


22 


24 


4 


28 


22 


2 15.0 


7 54 


24 


28 


20 


24 


25 


4 


29 


24 


2 23.0 


8 22 


26 


29 


21 


25 


27 


4 


31 


25 


2 30.9 


8 50 


27 


31 


22 


27 


28 


4 


33 


27 


2 38.9 


9 18 


28 


32 


24 


28 


30 


4 


35 


28 


2 46.8 


9 45 


30 


34 


25 


29 


31 


5 


37 


29 


2 54.8 


10 13 


31 


36 


26 


31 


33 


5 


38 


31 


3 2.7 


10 41 


33 


38 


27 


32 


34 


5 


40 


32 


3 10.6 


11 9 


34 


39 


28 


34 


36 


5 


42 


34 



6C 



TABLE XXXIX. 



Moorfs Motions for Minutes. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


10 


11 


12 


13 


14 


15 


16 


17 


18 


1 








1 















































2 





1 


1 








1 





1 
































3 


1 


1 


2 


1 


1 


I 





1 
































4 


1 


2 


3 


1 


1 


1 





1 
































5 


1 


2 


4 


1 


1 


1 





1 
































6 


1 


3 


4 


1 


1 


2 





2 
































7 


1 


3 


5 


1 


2 


2 





2 





. 


























8 


2 


4 


6 


2 


2 


2 





2 

















1 














9 


2 


4 


6 


2 


2 


2 





2 

















1 














10 


2 


5 


7 


2 


2 


3 





3 

















1 














11 


2 


5 


8 


2 


3 


3 





3 





1 





1 





1 














12 


2 


5 


9 


2 


3 


3 





3 





1 





1 





1 














13 


2 


6 


9 


3 


3 


3 


1 


4 





1 





1 





1 














14 


3 


6 


10 


3 


3 


4 


1 


4 





1 





1 





1 














15 


3 


7 


11 


3 


3 


4 


1 


4 





1 





1 





1 














16 


3 


7 


12 


3 


4 


4 


1 


4 





1 





1 





1 














17 


3 


8 


12 


3 


4 


4 


1 


5 





1 





1 





1 














18 


3 


8 


13 


4 


4 


5 


1 


5 





1 





1 





1 








1 





19 


4 


9 


14 


4 


4 


5 


1 


5 





1 





1 





1 








1 





20 


4 


9 


14 


4 


5 


5 


1 


5 





1 





1 





1 







1 





21 


4 


10 


15 


4 


5 


5 


1 


6 





1 





1 





1 







1 





22 


4 


10 


16 


4 


5 


6 


1 


6 





1 





1 


1 


2 







1 





23 


4 


10 


17 


5 


5 


6 


1 


6 





1 





1 


1 


2 







1 





'24 


5 


11 


17 


5 


6 


6 


1 


7 





1 




1 


1 


2 


1 




1 





25 


5 


11 


18 


5 


6 


6 


1 


7 





1 




1 


1 


2 


1 




1 





,26 


5 


12 


19 


5 


6 


7 


1 


7 





1 




1 


1 


2 


1 




1 


9. 


27 


5 


12 


19 


5 


6 


7 


1 


7 





1 




1 


1 


2 


1 




1 





28 


5 


13 


20 


6 


7 


7 


1 


8 





1 




1 


1 


2 


1 




1 





29 


6 


13 


21 


6 


7 


7 


1 


8 





1 




1 


1 


2 


1 




1 





30 


6 


14 


22 


6 


7 


8 


1 


8 





1 




1 


1 


2 


1 




1 






TABLE XXXIX. 



61 



Moorts Motions for Minutes. 











Sup. 


















Min. 


Evec. 


Anom. 


Varia. 


Long. 


Sod. 


II 


V 


VI 


VII 



vm 




IX 



XI 



XII 



1 


28 


32.7 


30 


32.9 


0.1 











2 


57 


1 5.3 


1 1 


1 5.9 


0.3 


1 























3 


1 25 


1 38.0 


1 31 


1 38.8 


0.4 


1 























4 


1 53 


2 10.6 


2 2 


2 11.8 


0.5 


2 























5 


2 2] 


2 43.3 


2 32 


2 44.7 


0.7 


2 























6 


2 50 


3 16.0 


3 3 


3 17.6 


0.8 


3 























7 


3 18 


3 48.6 


3 33 


3 50.6 


0.9 


3 























8 


3 46 


4 21.3 


4 4 


4 23.5 


1.1 


4 























9 


4 15 


4 54.0 


4 34 


4 56.5 


1.2 


4 























10 


4 43 


5 26.6 


5 5 


5 29.4 


1.3 


5 























11 


5 11 


5 59.3 


5 35 


6 2.4 


1.5 


5 























12 


5 40 


6 31.9 


6 6 


6 35.3 


1.6 


6 























13 


6 8 


7 4.6 


6 36 


7 8.2 


1.7 


6 























14 


6 36 


7 37.3 


7 7 


7 41.2 


1.9 


7 























15 


7 4 


8 9.9 


7 37 


8 14.1 


2.0 


7 























16 


7 33 


8 42.6 


8 8 


8 47.1 


2.1 


7 























17 


8 1 


9 15.3 


8 38 


9 20.0 


2.3 


8 























18 


8 29 


9 47.9 


9 9 


9 52.9 


2.4 


8 

















1 





19 


8 58 


10 20.6 


9 39 


10 25.9 


2.5 


9 

















1 





20 


9 26 


10 53.2 


10 10 


10 58.8 


2.6 


9 
















1 





21 


9 54 


11 25.9 


10 40 


11 31.8 


2.8 


10 
















1 





22 


10 22 


11 58.6 


11 11 


12 4.7 


2.9 


10 


1 












1 





23 


10 51 


12 31.2 


11 41 


12 37.6 


3.0 


11 


1 












1 





24 


11 19 


13 3.9 


12 12 


13 10.6 


3.2 


11 


1 











1 




25 


11 47 


13 36.6 


12 42 


13 43.5 


3.3 


12 


1. 











1 




26 


12 16 


14 9.2 


13 13 


14 16.5 


3.4 


12 


1 










1 




27 


12 44 


14 41.9 


13 43 


14 49.4 


3.6 


13 


1 










1 




28 


13 12 


15 14.6 


14 13 


15 22.3 


3.7 


13 


1 










1 




29 


13 40 


15 47.2 


14 44 


15 55.3 


3.8 


13 


1 










1 




30 


14 9 


16 19.9 


15 14 


16 28.2 


4.0 


14 


1 










1 





62 



TABLE XXXIX. 



Moon's Motions for Minutes. 





2 


3 


4 


5 


i. 


8 


9 


10 11 


12 


13 


14 


15 


16 


17 


18 


31 6 


14 


22 


6 


7 


8 1 


8 





1 


1 


1 


1 


2 


1 








32 6 


14 


23 


6 


7 


8 1 


9 


1 


2 


1 


2 


1 


2 


1 








33 6 


15 


24 


7 


8 


9 1 


9 


1 


2 


1 


2 


1 


2 


1 








34 6 


15 


25 


7 


8 


9 


1 


9 


1 


2 


1 


2 


1 


2 


1 








35 7 


16 


25 


7 


8 


9 


1 


10 


1 


2 


1 


2 


1 


2 


1 








36 7 


16 


26 


7 


8 


9 


1 


10 


1 


2 


1 


2 


1 


3 


1 








37 7 


17 


27 


7 


9 


10 


1 


10 


1 


2 


1 


2 


1 


3 


1 








38 7 


17 


27 


8 


9 


10 


2 


10 


1 


2 


1 


2 


1 


3 


1 








39 7 


18 


28 


8 


9 


10 


2 


11 


1 


2 


1 


2 


1 


3 


1 








40 8 


18 


29 


8 


9 


10 


2 


11 


1 


2 


1 


2 


1 


3 


1 








41 8 


19 


30 


8 


10 


11 


2 


11 


1 


2 


1 


2 


1 


3 


1 








42 8 


19 


30 


8 


10 


11 


2 


11 


1 


2 


1 


2 


1 


3 


1 








43 8 


19 


31 


9 


10 


11 


o 


12 


1 


2 


1 


2 


1 


3 


1 








44 8 


20 


32 


9 


10 


11 


2 


12 


1 


2 


1 


2 


1 


3 


1 








45 9 


20 


32 


9 


10 


12 


2 


12 


1 


2 


1 


2 


1 


3 


1 








46 9 


21 


33 


9 


11 


12 


2 


12 


1 


2 


1 


2 


1 


3 


1 








47 9 


21 


34 


9 


11 


12 


2 


13 


1 


2 


1 


2 


1 


3 


1 








48 9 


22 


35 


10 


11 


12 


2 


13 


1 


2 


1 


2 


1 


3 


1 








49 9 


22 


35 


10 


11 


13 


2 


13 


1 


2 


1 


2 


1 


3 


1 








50, 9 


23 


36 


10 


11 


13 


2 


13 


1 


2 


1 


2 


1 


3 


1 








51 10 


23 


37 


10 


12 


13 


2 


14 


1 


2 


1 


2 


1 


4 


1 








52 10 


24 


38 


10 


12 


13 


2 


14 


1 


3 


1 


3 


1 


4 


1 








53 10 


24 


38 


11 


12 


14 


2 


14 


1 


3 


1 


3 


1 


4 


1 








54 10 


24 


39 


11 


12 


14 


2 


14 


1 


3 


1 


3 


1 


4 


1 




2 




55 10 


25 


40 


11 


13 


14 


2 


15 


1 


3 


1 


3 


1 


4 


1 




2 




56 


11 


25 


40 


11 


13 


14 


2 


15 


1 


3 


1 


3 


1 


4 


1 




2 




57 


11 


26 


41 


11 


13 


15 


2 


15 


1 


3 


1 


3 


1 


4 


1 




2 




53 


11 


26 


42 


12 


13 


15 


2 


16 


1 


3 


1 


3 


1 


4 


1 


2 


2 




59 


11 


27 


43 


12 


14 


15 


2 


16 


1 


3 


1 


3 


1 


4 


1 


2 


2 


1 


60 


11 


27 


43.1 


12 


14 


15 | 


2 


16 


1 


3 


1 


3 


1 


4 


1 


2 


2 1 1 



TABLE XXXIX. 



63 



Moon's Motions for Minutes. 



Min. 


Evec. 


Anom. 


Varia. 


Long. 


Sup. 
Nod. 


II 


V 


VI 


VII 


vm 


IX 


XI 


XII 


31 


14 37 


16 52.5 


15 45 


17 1.2 


4.1 


14 


1 














32 


15 5 


17 25.2 


16 15 


17 34.1 


4.2 


15 


1 














33 


15 34 


17 57.9 


16 46 


13 7.1 


4.4 


15 


1 














34 


16 2 


18 30.5 


17 16 


18 40.0 


4.5 


16 


1 














35 


16 30 


19 3.2 


17 47 


19 12.9 


4.7 


16 


1 














36 


16 58 


19 35.8 


18 17 


19 45.9 


4.8 


17 


1 














37 


17 27 


20 8.5 


18 48 


20 18.8 


4.9 


17 


1 














38 


17 55 


20 41.2 


19 18 


20 51.8 


5.0 


18 


1 














39 


18 23 


21 13.8 


19 49 


21 24.7 


5.2 


18 


1 














40 


18 52 


21 46.5 


20 19 


21 57.6 


5.3 


19 


1 














41 


19 20 


22 19.2 


20 50 


22 30.6 


5.4 


19 


1 














42 


19 48 


22 51.8 


21 20 


23 3.5 


5.6 


20 


1 














43 


20 16 


23 24.5 


21 51 


23 36.5 


5.7 


20 


1 














44 


20 45 


23 57.1 


22 21 


24 9.4 


5.8 


21 


1 














45 


21 13 


24 29.8 


22 52 


24 42.3 


6.0 


21 


1 














46 


21 41 


25 2.5 


23 22 


25 15.3 


6.1 


21 


1 














47 


22 10 


25 35.1 


23 53 


25 48.2 


6.2 


22 


1 














48 


22 38 


26 7.8 


24 23 


26 21.2 


6.4 


22 


1 










1 




49 


23 6 


26 40.5 


24 54 


26 54.1 


6.5 


23 


1 














50 


23 34 


27 13.1 


25 24 


27 27.0 


6.6 


23 


1 














51 


24 3 


27 45.8 


25 55 


28 0.0 


6.8 


24 


1 














52 


24 31 


28 18.5 


26 25 


28 32.9 


6.9 


24 


1 














53 


24 59 


2S 51.1 


26 56 


29 5.9 


7.0 


25 


1 














54 


25 28 


29 23.8 


27 26 


29 38.8 


7.1 


25 


1 










2 




55 


25 56 


29 56.4 


27 56 


30 11.8 


7.3 


26 


1 










2 




56 


26 24 


30 29.1 


28 27 


30 44.7 


7.4 


26 


1 










2 




57 26 52 


31 1.8 


28 57 


31 17.6 


7.5 


27 


1 


2 








2 




58 27 21 


31 34.4 


29 28 


31 50 6 


7.7 


27 


1 


2 








2 




59 ! 27 49 


32 7.1 


29 58 


32 23.5 


7.8 


28 


1 


2 




i 




2 




60 1 28 17 


32 39.8 


SO 29 


32 56.5 


7.9 


28 


1 


2 




i! i 


2 





64 



TABLE XL. 



Moon's Motions for Seconds. 



Sec. 


Evec. 


Anom. 


Var. 


Long. 


Sec. 


Evec. 


Anom. 


Var. 


Long. 


1 





0.5 


1 


0.5 


31 


15 


16.9 


16 


17.0 


2 


1 


1.1 


1 


1.1 


32 


15 


17.4 


16 


17.6 


3 


1 


1.6 


2 


1.6 


33 


16 


18.0 


17 


18.1 


4 


2 


2.2 


2 


2.2 


34 


16 


18.5 


17 


18.7 


5 


2 


2.7 


3 


2.7 


35 


17 


19.1 


18 


19.2 


6 


3 


3.3 


3 


3.3 


36 


17 


19.6 


18 


19.8 


7 


3 


3.8 


4 


3.8 


37 


18 


20.1 


19 


20.3 


8 


4 


4.3 


4 


4.4 


38 


18 


20.7 


19 


20.9 


9 


4 


4.9 


5 


4.9 


39 


18 


21.2 


20 


21.4 


10 


5 


5.4 


5 


5.5 


40 


19 


21.8 


20 


22.0 


11 


5 


6.0 


6 


6.0 


41 


19 


22.3 


21 


22.5 


12 


6 


6.5 


6 


6.6 


42 


20 


22.9 


21 


23.1 


13 


6 


7.1 


7 


7.1 


43 


20 


23.4 


22 


23.6 


14 


7 


7.6 


7 


7.7 


44 


21 


24.0 


22 


24.2 


15 


7 


8.2 


8 


8.2 


45 


21 


24.5 


23 


24.7 


16 


8 


8.7 


8 


8.8 


46 


22 


25.0 


23 


25.3 


17 


8 


9.2 


9 


9.3 


47 


22 


25.6 


24 


25.8 


18 


9 


9.8 


9 


9.9 


48 


23 


26.1 


24 


26.4 


19 


9 


10.3 


10 


10.4 


49 


23 


26.7 


25 


26.9 


20 


9 


10.9 


10 


11.0 


50 


24 


27.2 


25 


27.4 


21 


10 


11.4 


11 


11.5 


51 


24 


27.8 


26 


28.0 


22 


10 


12.0 


11 


12.1 


52 


25 


28.3 


26 


28.5 


23 


11 


12.5 


12 


12.6 


53 


25 


28.9 


27 


29.1 


24 


11 


13.1 


12 


13.2 


54 


26 


29.4 


27 


29.6 


25 


12 


13.6 


13 


13.7 


55 


26 


29.9 


28 


30.2 


26 


12 


14.1 


13 


14.3 


56 


26 


30.5 


28 


30.7 


27 


13 


14.7 


14 


14.8 


57 


27 


31.0 


29 


31.3 


28 


13 


15.2 


14 


15 4 


58 


27 


31.6 


29 


31.8 


29 


14 


15.8 


15 


15.9 


5'} 


28 


32.1 


30 


32.4 


30 


14 


16.3 


15 


16.5 


60 


28 


32.7 


30 


32.9 



TAELE XLI. 
First Equation of Moon's Longitude. — Argument 1 



i Diff. j 
for 10 Am. 




5000 12 
5050,13 
510013 
5150! 13 
5200 14 
250 ! 14 

5300Jl4 
5350115 

5450|15 
5500 15 

oo50 16 
5600 16 
5650 16 
5700 17 
5750 17 

5800^7 
5S50 18 
590048 
5950 18 
6000 19 

6050 19 
6100 19 
6150,19 
6200 20 
20 



Diff. 
for 10 



2300 1 

2350 1 

2400 1 

2450 1 

2500 1 



6250 

6300 

6350 

6400 

6450'21 

.6500,21 

t2 IJ6550 21 

™ 6600 21 

%™ [6650 22 

t "^ -6700 22 

■;C750 1 22 

6800 22 
50 22 
,6900 22 
6950 22 
7000 23 

7050 23 
7100 23 
7150 23 
7200 23 
7250 23 

'300 23 
7350 23 
7400 23 
7450 23 
7500 23 



40.0 
0.3 

20.5 

40.7 
0.9 

20.9 

40.9 
0.8: 
20.5 
40.1 
59.6 

18.8 
37.SJ 
56.7 
15.3 
33.6 

51.6 

9.4 

26.9 

44.0 

0.8 

17.2 
33.3 
49.0 
4.2 
19.1 

33.5 

47.5 

1.0 

14.1 

26.6 

38.7 
50.3 

1 
11 

21.7 

31.1 
39.9 
48.1 
55. 

2. 

9.3 

15.2 
20.4 
25.0 
29.0 

32.3 

35.0 
37.1 
38.5 
39.3 



Arsr. 



Diff. 
for 10 



4.06 
4.04 
4.04 
4.04 
4.00 
4.00 

3.98 
3.94 
3.92 
3.90 
3.84 

3.80 
3.78 
3.72 
3.66 
3.60 

3.56 
3.50 
3.42 
3.36 
3.28 

3.22 
3.14 
3.04 
2.98 
2.88 

2.80 
2.70 
2.62 
2.50 

2.42 

2.32 
2.20 
2.10 
1.98 
1.88 

1.76 
1.64 
1.54 
1.42 
1.28 

1.18 
1.04 
0.92 
0.80 

0.66 

0.54 
0.42 
0.28 
0.16 



7500 23 
7550 23 
7600123 
7650J23 
770023 
7750 23 

7800j23 
7S50,23 
7900;23 
7950123 
S000|23 

8050J23 

8100122 
8150'22 
8200'22 
8250:22 



8300,22 
8350.22 
8400'22 
8450.21 
850021 

8550)21 

8600,21 
8650 21 
8700 20 
8750 20 

880020 
8850 20 
8900 19 
8950 19 
9000 19 

9050 18 
9100 18 
9150 18 
9200 IS 
9250 17 

9300 17 
9350 17 
940016 
9450,16 
950016 

9550J15 

9600 15 
9650 15 
9700 14 
9750 14 

980044 
9850 13 
9900 13 
9950,13 
10000 12 



39. 

39.4 

38.9 

37.7 

35.8 

33.3 

30.2 
26.4 
22.0 
16.9 

11.2 

4.9 
57.9 
50.3 
42.0 
33.2 

23.7 
13.7 
3.1 
51.9 
40.1 

27.8 
15.0 
1.6 
47.7 
33.3 



0.02 
0.10 
0.24 
0.38 
0.50 
0.62 

0.76 

0.88 
1.02 
1.14 
1.26 

1.40 
1.52 
1.66 
1.76 

1.90 

2.00 

2.12 
2.24 
2.36 
2 46 

2.56 

2.68 
2.78 
2.S8 



To 3 " 8 

3 16 

47 2 

U 3.34 



14.3 

57.2 
39.7 
21.8 
3.6| 
45.1 

26.2 

7.0 

47.6 

27.9 

7.9 

47.7 
27.4 
•6.8 
46.1 
25.3 

4.4 
43.4 
22.3 

1.2 
40.0 



3.42 

3.50 

3.58 
3.64 
3.70 
3.78 

3.84 
3.88 
3.94 
4.00 
4.04 

4.06 

4.12 

4.14 
4.16 
4.18 

4.20 

4.22 
4.22 
4.24 



66 TABLE XLII. 

Equations 2 to 7 of Moon's Longitude. Arguments 2 to 7 



Arg. 



2500 
2600 
2700 
2800 
2900 
3000 

3100 
3200 
3300 
3400 
3500 

3600 
3700 
3800 
3900 
4000 

4100 
4200 
4300 
4400 
4500 

4600 
4700 
4800 
4900 
5000 

5100 
5200 
5300 
5400 
5500 

15600 
5700 
5800 
'5900 
J6000 

6100 

:6200 

J6300 
6400 
6500 

6600 
6700 

6800 



457.3 
4 57.0 
4 56.1 
4 54.7 
4 52.7 
450.1 

447.0 
4 43.3 
4 39.1 
4 34.4 
4 29 2 

4 23 5 
417.4 
4 10.8 
4 3.9 
3 56.6 

3 48.9 
341.0 
3 32.7 
3 24.2 
3 15.5 

3 6.6 

2 57.6 

2 48.5 
2 39.2 
2 30.0 

2 20.8 
211.5 
2 2.4 
153.4 
144.5 

135.8 
127.3 
119.0 
1 11.1 
1 3.4 

56.1 

49:2 
42.6 
36.5 
30.8 

25.6 
20.9 
16.7 



69001013.0 
7000 9.9 



7100 
7200 
7300 
7400, 
7500 



7.3 
5.3 
3.9 
3.0 

2.7 



diff 



0.3 

0.9 
1.4 

2.0 
2.6 
3.1 
3.7 
4,2 
4.7 
5.2 
5.7 
6.1 
6.6 
6.9 
7.3 
7.7 
7.9 

3 
8.5 
8.7 
8.9 
9.0 
9.1 
9.3 
9.2 

2 
9.3 

1 
9.0 

.9 

.7 
8.5 
8.3 
7.9 
7.7 
7.3 
6.9 
6.6 
6.1 
5.7 
5.2 

47 
4.2 
7 
3.1 
2.6 
2.0 
1.4 
0.9 
0.3 



2.3 
2.4 
2.8 
3.3 
4,1 
5.1 

6.4 

7.8 
9.4 



011.3 
13.3 

15.5 
17.9 
20.5 
23.2 
26.1 

29.1 
32.2 
35.4 

38.8 
42.2 

45.7 
49.2 

52.8 

56.4 

1 0.0 

1 3.6 

1 7.2 
1 10.8 
1 14.3 

1 17.8 

121.2 
124.6 
127.8 
130.9 
133.9 

136.8 
139.5 
142.1 
144.5 
146.7 

148.7 
1 50.6 
152.2 
153.6 
154.9 

155.9 
156.7 
157.2 
157.6 
157.7 



diff 



0.1 

0.4 

0.5 
0.8 
1.0 
1.3 
1.4 
1.6 
1.9 
2.0 
2,2 
2.4 
2.6 
2.7 
2.9 
3.0 
3.1 
3.2 
3.4 
3.4 
3.5 

3.5 
3.6 

3.6 

.6 

3.6 

3.6 
3.6 
3.5 
3.5 
3.4 
3.4 
3.2 
3.1 
3.0 
2.9 
2.7 
2.6 
2.4 
2.2 
2.0 
1.9 
1.6 
1.4 
1.3 
1.0 
0.8 
05 
0.4 
0.1 



6 30.3 
6 29.9 
6 28.8 
6 26.9 
6 24.3 
6 21.0 

6 16.9 
6 12.2 
6 6 
6 
5 54.0 

5 46.6 
5 38.7 
5 30.3 
521.3 
5 11.9 

5 2.0 
451.7 

441.0 
4 30.1 

418.8 

4 7.3 
3 55.7 
3 43.9 
331.9 
3 20.0 

3 '8.1 
2 56.1 
2 44.3 
2 32.7 
221.2 

2 9.9 
1 59.0 
1 48.3 
138.0 
128.1 

118.7 
1 9.7 
1 1.3 
53.4 
46.0 

39.3 
33.2 
27.8 
23.1 
019.0 

015.7 
013.1 
011.2 
010.1 
9.7 



diff 



0.4 

1.1 

1.9 

2.6 

3.3 

4.1 

4.7 

5.4 

6.1 

6.7 

7.4 

7.9 

8.4 

9.0 

9.4 

9.9 

10.3 

10.7 

10.9 

11.3 

11.5 

11.6 

11.8 

12.0 

11.9 

11.9 

12.0 

11.8 

11.6 

11.5 

11.3 

10.9 

10.7 

10.3 

9.9 

9.4 

9.0 

8.4 

7.9 

7.4 

6.7 

6.1 
5.4 
4.7 
4.1 
3.3 

2.6 
1.9 
1.1 
0.4 



3 39.4 
3 39.2 
3 38.5 
3 37.5 
3 36.0 
3 34.1 

331.7 
3 29.0 
3 25.9 

3 22.4 
3 18.5 

314.3 
3 9.7 
3 4.9 
2 59.7 
2 54 3 

2 4S.6 
2 42.7 
2 36.6 
2 30.3 
2 23.8 

2 17.2 
2 10.5 
2 3.7 
156.9 
150.0 

143.1 

1 36.3 
129.5 
122.8 
1 16.2 

1 9.7 
1 3.4 
57.3 
51.4 
45.7 

40.3 
35.1 
30.3 
25.7 
021.5 

17.6 
014.1 
011.0 
8.3 
5.9 



diff 



0.2 
0.7 
1.0 
1.5 
1.9 
2.4 
27 
3.1 
3.5 
3.9 
4.2 
4.6 
4.8 
5.2 
5.4 
5.7 

5.9 

6.1 
6.3 
6.5 
6.6 
6.7 
6.8 
6.8 
.9 
6.9 

6.8 

6.8 
6.7 
6.6 
6.5 

6.3 
6.1 

5.9 
5.7 
5.4 
5.2 
4.8 
4.6 
4.2 
3.9 
3.5 
3.1 
2.7 
2.4 
1.9 
1.5 
1.0 
0.7 
0.2 








10.3 

012.1 
14.2 
16.6 
19.2 
22.2 

25.4 
28.9 
32.7 
36.6 
40.7 

45.1 
49.6 
54.3 

59.2 

1 4.1 

1 9.2 
1 14.3 
1 19.5 
124.7 
130.0 

135.3 

1 40.5 
145.7 
150.8 
155.9 

i* 08 

2 5.7 
210.4 
214.9 
2193 

2 23.4 
2 27.3 
231.1 
2 34.6 

2 37.8 

2 40.8 
2 43.4 
2 45.8 
2 47.9 
2 49.7 

2 51.2 
2 52.3 
2 53.1 
2 53.6 
2 53.8 



diff 



0.2 
0.5 
0.8 
1.1 
1.5 
1.8 
2.1 
2.4 
2.6 
3.0 
3.2 
3.5 
3.8 
3.9 
4.1 
4.4 
4.5 
4.7 
4.9 
4,9 
5.1 
5.1 
5.2 
5.2 
5.3 
5.3 

5.2 
5.2 
5.1 
5.1 
4.9 

4.9 
4.7 
4,5 
4.4 
1,1 
3.9 
3.8 
3.5 
3.2 
3.0 

2.6 
2.4 
2.1 
1.8 
1.5 

1.1 

0.8 
0.5 
0.2 



0.8 
0.9 
1.3 
1.8 
2.7 
3.7 



5.0 
6.4 

8.1 









10.0 

12.1 

014.4 
16.8 
19.5 
22.3 
25.2 

28.3 
031 
34.8 
38.2 
041.7 

45.3 
048.9 
52 

56.3 

1 0.0 

1 3.7 

1 7.4 
1 11.1 
1 14.7 
1 18 3 

1218 

125.2 
128.5 
1 31.7 
134.8 

137.7 
140.5 
143.2 
145.6 
1 47.9 

1 50.0 
151.9 
1 53.6 
155.0 
156.3 

1 57.3 
1 58.2 
158.7 
159.1 
159.2 



cliff 



0.1 
0.4 
0.5 
0.9 
1.0 
1.3 
1.4 
1.7 
1.9 
2.1 
2.3 
2.4 
2.7 
2.8 
2.9 
3.1 
3.2 
3.3 
3.4 
3.5 
3.6 
3.6 
3.7 
3.7 
3.7 
3.7 

3.7 
3.7 
3.6 
3.6 
3.5 
3.4 
3.3 
3.2 
3.1 
2.9 

2.8 
2.7 
2.4 
2.3 
2.1 

1.9 
1.7 

4 
1.3 


0.9 
0.5 
0.4 
0.1 



Arg. 



2500 
2400 
2300 
2200 
2100 
2000 

1900 
1800 
1700 
1600 
1500 

1400 
1300 
1200 
1100 
1000 

900 
800 
700 
600 
500 

400 
300 
200 
100 


9900 
9800 
9700 
9600 
9500 

9400 
9300 
9200 
9100 
9000 

8900 
8800 
8700 
8600 
8500 

8400 
8300 
8200 
8100 
8000 

7900 
7800 
7700 
7600 
7500 



TABLE XLIII. 



TABLE XLIV, 



G7 



Equations 8 and 9. 



Equations 10 and 11. 



Arg. 



S 


9 | 


Arg. 


8 


9 
1 20.0 


r 


^g. 



10 
LO.O 


10.0] 


Arg. 
500 


10 
10.0 


11 


1 20.0 


1 20 Oj 5000 


1 20.0 


10.0 


100 


1 15.5 


1 28 7 { 5100 


1 24.4 


1 25.8 




10 


9.3 


11.1 


510 


9.6 


10.8 


200 


1 11.1 


1 37.3 j 5200 


1 28.8 


1 31.4 




20 


8.6 


12.1 : 


520 


9.2 


11.5 


300 


1 6.7 


1 45.7 6 5300 


1 33.1 


1 36.9 




30 


8.0 


13.1 j 


530 


8.9 


12.3 


400 


1 2.3 


1 53.7 1 5400 


1 37.4 


1 42.0 




40 


7.4 


14.1 ; 


540 


8.5 


12.9 


500 


58.0 


2 1.3 


5500 


1 41.6 


1 46.8 




50 


6.8 


15.0 '' 


550 


8.2 


13.6 


600 


53.8 


2 8.3 


5600 


1 45.8 


1 51.0 




60 


6.2 


15.8 


560 


7.9 


14.2 


700 


49.7 


2 14.7 


5700 


1 49.8 


1 54.6 




70 


5.7 


16.6 


570 


7.7 


14.6 


800 


45.7 


2 20.2 


5800 


1 53.8 


1 57.6 




80 


5.3 


17.3 


580 


7.5 


15.0 


900 


41.9 


2 25.0 


5900 


1 57.6 


1 59.8 




90 


4.9 


17.9; 


590 


7.4 


15.4 


1000 


38.2 


2 28.9 


6000 


2 1.2 


2 1.3 




100 


4.6 


18.3 


600 


7.3 


15.6 


1100 


34.7 


2 31.92 6100 


2 4.7 


2 1.9 




110 


4.3 


18.6. 


610 


7.2 


15.7 


1200 


31.4 


2 33.9 


6200 


2 8.0 


2 1.7 




120 


4.1 


18.9 ; 


620 


7.3 


15.7 


1300 


28.2 


2 34.9 


6300 


2 11.2 


2 0.7 




130 


4.0 


19.0 


630 


7.4 


15.6 


1400 


25.3 


2 35.0 


6400 


2 14.1 


1 58.8 




140 


4.0 


18.9 


640 


7.5 


15.4 


1500 


22.6 


2 34.1 


6500 


2 16.8 


1 56.1 




150 


4.0 


18.8 '. 


650 


7.8 


15.1 


1600 


20.1 


2 32.2 


6600 


2 19.3 


1 52.5 




160 


4.2 


18.6 


660 


8.1 


14.7 


1700 


17.9 


2 29.5 


6700 


2 21.6 


1 48.3 




170 


4.4 


18.2 


670 


8.4 


14.2 


1800 


15.9 


2 25.9 


6800 


2 23.7 


1 43.4 




180 


4.6 


17.7 


680 


8.7 


13.5 


1900 


14.2 


2 21.5 


6900 


2 25.4 


1 37.8 




190 


4.9 


17.1 


690 


9.2 


12.8 


2000 


12.7 


2 16.4 


7000 


2 27.0 


1 31.7 




200 


5.3 


16.5 


700 


9.7 


12.1 


2100 


11.5 


2 10.71 7100 


2 28.2 


1 25.1 




210 


5.7 


15.7 


710 


10.2 


11.3 


2200 


10.5 


2 4.4 1 7200 


2 29.2 


1 18.2 




220 


6.2 


14.9 


720 


10.7 


10.4 


2300 


9.9 


1 57.71 7300 


2 30.0 


1 11.1 




230 


6.7 


14.1 


730 


11.2 


9.5 


2400 


9.5 


1 50.7 1 7400 


2 30.4 


1 3.8 




240 


J.2 


13.2 


740 


11.7 


8.6 


2500 


9.4 


1 43.5 7500 


2 30.6 


56.5 




250 


7.7 


12.3 


750 


12.3 


7.7 


2600 


9.6 


1 36.2 


7600 


2 30.5 


49.3 




260 


8.3 


11.4 


760 


12.8 


6.8 


2700 


10.1 


1 28.9 


7700 


2 30.1 


42.3 




270 


8.8 


10.5 


770 


13.3 


5.9 


2800 


10.8 


1 21.8 


7800 


2 29.5 


35.6 




2S0 


9.3 


9.6 


780 


13.8 


5.1 


2900 


11.8 


1 14.9 


7900 


2 28.5 


29.3 




290 


9.8 


8.7 


j 790 


14.3 


4.3 


3000 


13.0 


1 8.3 


80Q0 


2 27.3 


23.6 




300 


10.3 


7.9 


1 800 


14.7 


3.5 


3100 


14.6 


1 2.2 


8100 


2 25.8 


18.5 




310 


10.8 


7.2 


810 


15.1 


2.9 


3200 


16.3 


56.6 


i 8200 


2 24.1 


14.1 




320 


11.3 


6.5 


820 


15.4 


2.3 


3300 


18.4 


51.7 


8300 


2 22.1 


10.5 




330 


11.6 


5.8 


i 830 


15.6 


1.8 


3400 


20.7 


47.5 


8400 


2 19.9 


7.8 




340 


11.9 


5.3 


: 840 


15.8 


1.4 


3500 


23.2 


43.9 


8500 


2 17.4 


5.9 




350 


12.2 


4.9 


| 850 


16.0 


1.2 


3600 


25.9 


41.2 


! 8600 


2 14.7 


5.0 




360 


12.5 


4.6 


"i 860 


16.0 


1.1 


3700 


28.8 


39.3 


I 8700 

' 8800 

8900 


2 11.8 


5.1 




370 


12.6 


4,4 


! 870 


16.0 


1.0 


3800 


32.0 


38.3 


12 8.6 


6.1 




3S0 


12.7 


4.3 


! 880 


15.9 


1.1 


3900 


35.3 


38.1 


2 5.3 


8.1 




390 


12.8 


4.3 


] 890 


15.7 


1.4 


4000 


38.8 


38.7 


9000 


2 1.8 


11.1 




400 


12.7 


4.4 


j 900 


15.4 


1.7 


4100 


42.4 


40.2 


9100 


1 5S.1 


15.0 




410 


12.6 


4.6 


j 910 


15.1 


2.1 


4200 


46.2 


42.4 


9200 


1 54.3 


19.8 




420 


12.5 


5.0; 920 


14.7 


2.7 


4300 


50.2 


45.4 


9300 


1 50.3 


25.3 




430 


12.3 


5.4 i 930 


14.3 


3.4 


4400 


54.2 


49.0i 9400 


1 46.2 


31.7 




440 


12.1 


5.8 ! 940 


13.8 


4.2 


4500 


58.4 


53.2 § 9500 


1 42.0 


38.7 




450 


11.8 


6.4 < 950 


13.2 


5.0 


4600 


1 2.6 


58.0 B 9600 


1 37.7 


46.3 




460 


11.5 


7.1 ! 960 


12.6' 5.9 


4700 


1 6.9 


1 3.11 9700 


1 33.3 


54.3 




470 


11.1 


7.7 1 970 


12.0 ' 6.9 


4800 


1 11.2 


1 8.61 9800 


1 28.9 


1 2.7 




480 


10.8 


8.5 ! 980 


lL4i 7.9 


4900 


1 15.6 


1 14.21 9900 


1 24.5 


1 11.3 




490 


10.4 


9.2 ] 9L0 


10.7 8.9 


5000 


1 20.0 


1 20.0 1 10000 11 20.0 


1 20.0 




500 


10.0 


10.0 *10( 


10.0 1 10.0 

J 



68 



TABLE XLT. 
Equations 12 to 19- 



TABLE XLVL 
Equation 20. 



Arg. 


12 


13 


Ji 


15 


16 


17 


18 1 19 


Arg. 
250 


250 


2.3 


1.6 


7.8 


0.0 


33.7 


3.4 


16.7 


0.4 


260 


2.3 


1.6 


7.8 


0.0 


33.7 


3.4 


16.7 


0.4 


240 


270 


2.4 


1.7 


7.9 


0.1 


33.6 


3.5 


16.6 


0.4 


230 


280 


2.6 


1.9 


8.0 


0.2 


33.5 


3.5 


16.6 


0.5 


220 


290 


2.9 


2.2 


8.2 


0.3 


33.2 


3.6 


16.5 


0.5 


210 


300 


3.2 


2.5 


8.4 


0.5 


33.0 


3.7 


16.4 


0.6 


200 


310 


3.5 


2.9 


8.7 


0.7 


32.7 


3.9 


16.2 


0.7 


190 


320 


4.0 


3.4 


&.0 


1.0 


32.4 


4.0 


16.1 


0.8 


180 


330 


4.5 


3.9 


9.3 


1.2 


32.0 


4.2 


15.9 


1.0 


170 


340 


5.1 


44 


9 7 


1 6 


31.6 


4.4 


15.7 


1.1 


160 


350 


5.7 


5.1 


10.1 


1.9 


31.1 


4.7 


15.4 


1.3 


150 


360 


6.4 


5.8 


10.6 


2.3 


30.6 


4.9 


15.2 


1.5 


140 


370 


7.1 


6.6 


11. 1 


2.7 


30.1 


5.2 


14.9 


1.7 


130 


380 


7.9 


7.4 


11.7 


3.2 


29.4 


5.5 


14.6 


1.9 


120 


390 


8.7 


8.3 


12.2 


3.6 


28.7 


5.8 


14.3 


2.1 


110 


400 


9.6 


9.2 


12.8 


4.1 


28.0 


6.1 


13.9 


2.3 


100 


410 


10.5 


10.1 


13.5 


4.6 


27.3 


6.5 


13.6 


2.5 


90 


420 


11.5 


11.1 


14.1 


5.2 


26.6 


6.8 


13.2 


2.8 


80 


430 


12.5 


12.2 


14.8 


5.7 


25.8 


7.2 


12.9 


3.1 


70 


440 


13.5 


13.2 


15.5 


6.3 


25.0 


7.6 


12.5 


3.3 


60 


450 


14.5 


14.3 


16.2 


6.9 


24.2 


8.0 


12.1 


3.6 


50 


460 


15.6 


15.4 


17.0 


7.5 


23.4 


8.4 


11.7 


3.9 


40 


470 


16.7 


16.5 


17.7 


8.1 


22.6 


8.8 


11.3 


4.1 


30 


480 


17.8 


17.7 


18.5 


8.7 


21.7 


9.2 


10.8 


4.4 


20 


490 


18.9 


18.8 


19.2 


9.4 


20.9 


9.6 


10.4 


4.7 


10 


500 


20.0 


20.0 


20.0 


10.0 


20.0 


10.0 


10.0 


5.0 





510 


21.1 


21.2 


20.8 


10.6 


19.1 


10.4 


9.6 


5.3 


990 


520 


22.2 


22.3 


21.5 


11.3 


18.3 


10.8 


9.2 


5.6 


980 


530 


23.3 


23.5 


22.3 


11.9 


17.4 


11.2 


8.7 


5.9 


970 


540 


24.4 


24.6 


23.0 


12.5 


16.6 


11.6 


8.3 


1 6.1 


960 


550 


25.5 


25.7 


23.8 


13.1 


15.8 


12.0 


7.9 


6.4 


950 


560 


26.5 


26.8 


24.5 


13.7 


15.0 


12.4 


7.5 


6.7 


910 


570 


27.5 


27.8 


25.2 


14.3 


14.2 


12.8 


7.1 


6.9 


930 


580 


28.5 


28.9 


25.9 


14.8 


13.4 


13.2 


6.8 


7.2 


920 


590 


29.5 


29.9 


26.5 


15.4 


12.7 


13.5 


6.4 


7.5 


910 


600 


30.4 


30.8 


27.2 


15.9 


12.0 


13.9 


6.1 


7.7 


900 


610 


31.3 


31.7 


27.8 


16.4 


11.3 


14.2 


5.7 


7.9 


890 


620 


32.1 


32.6 


28.3 


16.8 


10.6 


14.5 


5.4 


8.1 


880 


630 


32.9 


33.4 


28.9 


17.3 


9.9 


14.8 


5.1 


8.3 


870 


640 


33.6 


34.2 


29.4 


17.7 


9.4 


15.1 


4.8 


8.5 


860 


650 


34.3 


34.9 


29.9 


18.1 


8.9 


15.3 


4.6 


8.7 


850 


660 


34.9 


35.6 


30.3 


18.4 


8.4 


15.6 


4.3 


8.9 


840 


670 


35.5 


36.1 


30.7 


18.8 


8.0 


15.8 


4.1 


9.0 


830 


680 


36.0 


36.6 


31.0 


19.0 


7.6 


16.0 


3.9 


9.2 


820 


690 


36.5 


37.1 


31.3 


19.3 


7.3 


16.1 


3.8 


9.3 


810 


700 


36.8 


37.5 


31.6 


19.5 


7.0 


16.3 


3.6 


9.4 


800 


710 


37.1 


37.8 


31.8 


19.7 


6.8 


16.4 


3.5 


9.5 


790 


720 


37.4 


38.1 


32.0 


19.8 


6.5 


16.5 3.4 


9.5 


780 


730 


37.6 


38.3 


32.1 


19.9 


6.4 


16.5 3.4 


9.6 


770 


740 


37.7 38.4 


32.2 


20.0 


6.3 


16.6 3.3 


9.6 


760 


750 


37.7 38.4 


32.2 


20.0 


6.3 


16.6 3.3 


9.6 


750 



Arg. 


20 


Arg. 





10.0 


500 


10 


10.9 


5ia 


20 


11.8 


520 


30 


12.7 


530 


40 


13.5 


540 


50 


14.3 


550 


60 


15.0 


560 


70 


15.7 


570 


80 


16.2 


580 


90 


16.7 


590 


100 


17.0 


600 


110 


17.2 


610 


120 


17.4 


620 


130 


17.4 


630 


140 


17.2 


640 


150 


17.0 


650 


160 


16.7 


660 


170 


16.2 


670 


180 


15.7 


680 


190 


15.0 


690 


200 


14.3 


700 


210 


13.5 


710 


220 


12.7 


720 


230 


11.8 


730 


240 


10.9 


740 


250 


10.0 


750 


260 


9.1 


760 


270 


8.2 


770 


280 


7.3 


780 


290 


6.5 


790 


300 


5.7 


800 


310 


5.0 


810 


320 


4.3 


820 


330 


3.8 


830 


340 


3.3 


840 


350 


3.0 


850 


360 


2.8 


860 


370 


2.6 


870 


380 


2.6 


880 


390 


2.8 


890 


400 


3.0 


900 


410 


3.3 


910 


420 


3.8 


920 


430 


4.3 


930 


440 


5.0 


940 


450 


5.7 


950 


460 


6.5 


960 


470 


7.3 


970 


480 


8.2 


980 


490 


9.1 


990 


] 500 


10.0 


lOW 



TABLE XLVII. 



TABLE XLVIII. 69 



Equations 21 to 29. 



Equations 30 and 31. 



> 


21 


22 


23 


24 


25 


26 


27 


28 


29 


£ 




„ 


„ 


„ 


// 


// 


„ 


,/ 


„ 


,/ 




25 


7.8 


3.2 


7.1 


6.1 


5.9 


4.1 


5.8 


4.3 


5.7 


25 


27 


7.8 


3.2 


7.1 


6.1 


5.9 


4.1 


5.8 


4.3 


5.7 


23 


29 


7.7 


3.3 


7.0 


6.1 


5.9 


4.1 


5.8 


4.3 


5.7 


21 


31 


7.6 


3.3 


7.0 


6.0 


5.8 


4.2 


5.7 


4.3 


5.7 


19 


33 


7.5 


3.4 


6.8 


6.0 


5.8 


4.2 


5.7 


4.4 


5.6 


17 


35 


7.3 


3.5 


6.7 


5.9 


5.7 


4.3 


5.6 


4.4 


5.6 


15 


37 


7.0 


3.7 


6.5 


5.8 


5.7 


4.3 


5.6 


4.5 


5.5 


13 


39 


6.8 


3.9 


6.3 


5.7 


5.6 


4.4 


5.5 


4.6 


5.4 


11 


41 


6.5 


4.0 


6.1 


5.6 


5.5 


4.5 


5.4 


4.6 


5.4 


09 


43 


6.2 


4.2 


5.9 


5.5 


5.4 


4.6 


5.3 


4.7 


5.3 


07 


45 


5.9 


4.4 


5.6 


5.3 


5.3 


4.7 


5.2 


4.8 


5.2 


05 


47 


5.5 


4.7 


5.4 


5.2 


5.2 


4.8 


5.1 


4.9 


5.1 


03 


49 


5.2 


4.9 


5.1 5.1 


5.1 


4.9 


5.0 


5.0 


5.0 


01 


51 


4.8 


5.1 


4.9 


4.9 


4.9 


5.1 


5.0 


5.0 


5.0 


99 


53 


4.5 


5.3 


4.6 


4.8 


4.8 


5.2 


4.9 


5.1 


4.9 


97 


55 


4.1 


5.6 


4.4 


4.7 


4.7 


5.3 


4.8 


5.2 


4.8 


95 


57 


3.8 


5.8 


4.1 


4.5 


4.6 


5.4 


4.7 


5.3 


4.7 


93 


59 


3.5 I 


6.0 


3.9 


4.4 


4.5 


5.5 


4.6 


5.4 


4.6 


91 


61 


3.2 I 


6.1 


3.7 


4.3 


4.4 


5.6 


4.5 


5.4 


4.6 


89 


63 


3.0 j 


6.3 


3.5 


4.2 


4.3 


5.7 


4.4 


5.5 


4.5 


87 


65 


2.7 


6.5 


3.3 


4.1 


4.3 


5.7 


4.4 


5.6 


4.4 


85 


67 


2.5 


6.6 


3.2 


4.0 


4.2 


5.8 


4.3 


5.6 


4.4 


83 


69 


2.4 


6.7 


3.0 


4.0 


4.2 


5.8 


4.3 


5.7 


4.3 


81 


71 


2.3 


6.7 


3.0 


3.9 


4.1 


5.9 


4.2 


5.7, 


4.3 


79 


73 


2.2 


6.8 


2.9 


3.9 


4.1 


5.9 


4.2 


5.7] 


4.3 


77 


75 


2.2 


6.8 


2.9 


3.9 


4.1 


5.9 


4.2 


5.7 l 


4.3 1 


75 



TABLE XLIX. 
Equation 32. Argument, Supp. of Node. 





Ills 


TVs 


Ys 


Vis 


VII* 


VIII* 




o 



3.1 


4.0 


6.5 


10.0 


13.5 


16.0 


o 

30 


2 


3.1 


4.2 


6.8 


10.2 


13.7 


16.1 


28 


4 


3.1 


4.3 


7.0 


10.5 


13.8 


16.2 


26 


6 


3.1 


4.4 


7.2 


10.7 


14.0 


16.3 


24 


8 


3.2 


4.6 


7.4 


11.0 


14.2 


16.4 


22 


10 


3.2 


4.7 


7.6 


11.2 


14.4 


16.5 


20 


12 


3.3 


4.9 


7.9 


11.4 


14.6 


16.6 


18 


14 


3.3 


5.0 


8.1 


11.7 


14.8 


16.6 


16 


16 


3.4 


5.2 


8.3 


11.9 


15.0 


16.7 


14 


18 


3.4 


5.4 


8.6 


12.1 


15.1 


16.7 


12 


20 


3.5 


5.6 


8.8 


12.4 


15.3 


16.8 


10 


22 


3.6 


5.8 


9.0 


12.6 


15.4 


16.8 


8 


24 


3.7 


6.0 


9.3 


12.8 


15.6 


16.9 


6 


26 


3.8 


6.2 


9.5 


13.0 


15.7 


16.9 


4 


28 


3.9 


6.3 


9.8 


13.2 


15.8 


16.9 


2 


30 


4.0 


6.5 


10.0 


13.5 


16.0 


16.9 







II* 


I* 


o* 


XI* 


X* 


IX* 





Arg. 





2 
4 
6 

S 

10 
12 
14 
16 
18 

20 

22 
24 
26 
28 

30 
32 
34 
36 
38 

40 
42 
44 
46 

48 

50 
52 
54 
56 
58 

60 

62 

64 
63 

68 

70 

72 
74 

76 

78 

80 
82 
84 
86 
88 

90 
92 
94 
96 
98 
100 



30 



5.0 
5.0 
4.9 
4.9 
4.8 

4.8 
4.7 
4.6 
4.5 
4.4 

4.2 

4.1 
4.0 
3.9 
3.8 

3.7 
3.7 
3.7 
3.7 
3.8 

3.9 
4.1 
4.3 
4.5 

4.8 

5.0 
5.2 
5.5 
5,7 
5.9 

6.1 
6.2 
6.3 
6.3 
6.3 

6.3 

6.2 
6.2 
6.0 
5.9 

5.8 
5.7 
5.5 
5.4 
5.3 

5.2 
5.1 
5.1 
5.0 
5.0 
5.0 



31 



5.0 
5.0 
5.1 
5.1 

5.2 

5.2 
5.3 
5.4 
5.5 
5.5 

5.6 

5.7 
5.8 
5.8 
5.9 

5.9 

5.9 
5.9 
5.9 
5.8 

5.7 
5.6 
5.5 
5.3 
5.2 

5.0 

4.8 
4.7 
4.5 
4.4 

4.3 
4.2 
4.1 
4.1 
4.1 

4.1 
4.1 
4.2 
4.2 
4.3 

4.4 
4.5 
4.6 
4.6 
4.7 

4.8 
4.8 
4.9 
4.9 
5.0 
5.0 



Constant 55" 



70 



TABLE L. 

Evection. 



Argument. Evection, corrected. 



O 



lis 



Ills 



lYs 



30 00.0 

31 25.5 

32 50.9 

34 16.3 

35 41.6 

37 6.7 

38 31.8 

39 56.7 

41 21.4 

42 45.8 
44 10.1 1 



Diff. 2o 



85.5 
85.4 
85:4 
S5.3 
85.1 

85.1 

84.9 

84.7 
84.4 
84.3 

83.9 



Diff. 2 : 



Diff. 



10 43.5 

11 56.7! 



73.2 



^X~S3.7 
46 57.7 oo A 

48 21.1 ^ 

49 44.1 so n 
51 6.7 8 - 6 

,82.2 

16 52 28.9 qi R 

17 53 50.7 q: "« 

18 55 12.0 ^ 
19,56 32.9ftn'o 
20 57 53.2 8 °- 3 

179.8 

5 A_ 13 : 79.3 

32.3 7Q 7 

1 51.0 73'! 

3 9.1 77.4 

4 26.5 
76.8 

76.1 



5 43.3 

6 59.4 

8 14.9 

9 29.6 
10 43.5 



75.5 
74.7 
73.9 



13 9.0 -g 

14 20.6 ^ 

15 31.3 fi n o 

16 41.ir yb 
69.0 

68.1 
67.1 
66.2 
65.2 

64.3 
63.2 
62.2 
61.2 
60.0 

59.0 

58.0 
56.8 
55.6 
54.5 

53.3 

52.2 

50:9 

49.8 
48.5 

! 47.2 

46.0 
44.8 
43.4 
42.1 



17 50.1 

18 58.2 

20 5.3 

21 11.5 

22 16.7i 

23 21.0 

24 24.2 

25 26.4 

26 27.6 

27 27.6, 

28 26.6 

29 24.6 
130 21.4 

31 17.0 

32 11.5 

33 4.8 

33 57.0 

34 47.9 

35 37.7 

36 26.2 

37 13.4 

37 59.4 

38 44.2 

39 27.6 

40 9.7 



40 9.7 

40 50.6 

41 30.1 

42 8.3 
42 45.1 
j 43 20.6 

'43 54.7 

44 27.4 

44 58.8 

45 28.7 

45 57.3 

! 46 24.5 

46 50.2 

47 14.5 
;47 37.4 
! 47 58.8 

l4S 18.8 
|48 37.4 

48 54.5: 
|49 10.1 j 

49 24.4 

49 37.1 

49 48.3 

49 58.1 

50 6.4 
50 13.3 

50 1S.7 
50 22.6 
50 25.0 
50 26.0 
50 25.5 



40.9 
39.5 
38.2 
|36.8 
35.5 

34.1 

32.7 
31.4 

29.9 
[28.6 

27.2 

25.7 
24.3 
22.9 
21.4 

20.0 

18.6 
17.1 
15.6 
14.3 

12.7 

11.2 
9.8 
8.3 
6.9 

5.4 

3.9 
2.4 
1.0 
0.5 



Diff. 



2P 



Diff. 2^ 



Diff. 



50 25.5 
50 23.5 
50 20.1 
50 15.2 
50 8.8 
50 1.0 

49 51.7 
49 41.0 
49 28.8 
49 15.1 
49 0.2 

48 43.5 
48 25.6 
48 6.3 
47 45.5 
47 23.3 

46 59.8 
46 34.8 
46 8.5 
45 40.7 
45 11.6 

44 41.2 
44 9.5 
43 36.4 
43 1.9 1 
42 26.2 



2.0 
3.4 

4.9 
6.4 

7.8 

9.3 

10.7 
12.2 
13.7 
14.9 

16.7 
17.9 
19.3 

20.8 
22.2 

23.5 
25.0 
26.3 

27.8 
29.1 

30.4 

31.7 
33.1 
34.5 
35.7 



41 49.2 
41 10.8 
40 31.2 
39 50.4 
39 8.3 

J2° 



37.0 

38.4 
39.6 
40.8 
42.1 



39 8.3 
38 24.9 
37 40.4 
36 54.6 
36 7.6 
35 19.5 

34 30.2 
33 39.7 
32 48.1 
31 55.4 
31 1.6 

30 6.7 
29 10.7 
28 13.7 
27 15.7 
26 16.6 

25 16.6 
24 15.6 
23 13.6 
22 10.7 
21 6.8 

20 2.1 
18 56.4 
17 49.9 
16 42.6 
15 34.4 



14 25.5 
13 15.7 
12 5.2 
10 54.0 
9 42.0 



43.4 
44.5 
45.8 
47.0 
48.1 

49.3 
50.5 
51.6 

52.7 
53.8 

54.9 
56.0 
57.0 
58.0 
59.1 

60.0 
61.0 



9 42.0 
8 29.3 
7 16.0 
6 2.0 
4 47.4 
3 32.2 

2 16.3 
59.9 



59 43.0 
58 25.6 
57 7.6 

55 49.2 
54 30.3 
53 11.0 
51 51.3 
50 31.2 



49 10.71 

A 47 49.9 

S2'2 46 2S.8 

45 7.5 

43 45.8 



62.9 
63.9 



64.7 

65.7 
66.5 
67.3 
68.2 



68.9 

69.8 
70.5 
71.2 
72.0 



42 23.9 

41 1.8 

39 39.5 

38 17.0 

36 54.4 
I 

35 31.7 
34 8.8 
32 45.91 
31 23.0 
30 0.0 

1° 



72.7 
73.3 
74.0 
74.6 
75.2 

75.9 

76.4 
76.9 

77.4 
78.0 

78.4 

78.9 f 
79.3 
79.7 
80.1 

80.5 

80.8 
81.1 
81.3 
81.7 

81.9 

82.1 
82.3 
82.5 
82.6 

82.7 

82.9 
82.9 
82.9 
83.0 



TABLE L. 

Evection. 



71 



Argument. Evection, corrected. 



VI* 



VII* 



VIII* 



IX* 



Xs 



XI* 



«1< 



30 
128 
2 27 
325 
424 
5 ! 23 



Diff.0 c 



16 

17 6 

18 5 

19 4 

20 2 

21 1 

22 _0 

23 59 

24 57 

25 56 

12655 

27 53 

28 52 
29,51 
30 50 



00 
37.0 
14.1 
51.2 

28.3 
5.6 

43.0 
20.5 
58.2 
36.1 
14.2 

52.5 
31.2 
10.1 
49.3 

28.8 

8.7 
49.0 
29.7 
10.8 
52.4 

34.4 

17.0 

0.1 

43.7 

27.S 

12.6 
58.0 
44.0 
30.7 
18.0 



83.0 

82.9 
S2.9 
S2.9 
82.7 

82.6 
82.5 
82.3 
82.1 
81.9 

81.7 

81.3 
81.1 

80.8 
80.5 

80.1 

79.7 
79.3 

78.9 
78.4 

78.0 

77.4 
76.9 
76.4 
75.9 

75.2 

74.6 
74.0 
i73.3 

72.7 
I 



50 18.0 
49 6.0 
47 54.8 
46 44.3 
45 34.5 
44 2*5.6 

43 17.4 
42 10.1 
41 3.6 
39 57.9 
38 53.2 

37 49.3 
36 46'.4 
35 44.4 
34 43.4 
33 43.4 

32 44.3 
31 46.3 
30 49.3 
29 53.3 

28 58.4 

28 4.6 
27 11.9 
26 20.3 
25 29.8 
24 40.5 

23 52.4 
23 5.4 
22 19.6 
21 35.1 
20 51.7 

0° 



Diff.0 c 



Diff.O Diff. 0^ 



72.0 
71.2 
70.5 
69.8 
68.9 

68.2 

67.3 

66.5 
65.7 
64.7 

63.9 
62.9 
62.0 
61.0 
60.0 

59.1 

58.0 
57.0 
56.0 
54.9 

53.8 

52.7 
51.6 
50.5 
49.3 

48.1 

47.0 
45.8 
44.5 
43.4 



20 51 
20 9 
19 28 
18 49 
18 10 
17 33 

16 58 
16 23 
15 50 
15 18. 
14 48. 

14 19, 
13 51. 
13 25. 
13 0. 
12 36. 

12 14. 
11 53, 
11 34 
11 16 
10 59 

10 44 
10 31 
10 19 
10 8 
9 59 

9 51 
9 44 
9 39 
9 36 
9 34 

0° 



42.1 
40.8 
39.6 
38.4 
37.0 

35.7 
34.5 
33.1 
31.7 
30.4 

29.1 
27.8 
26.3 
25.0 
23.5 

22.2 

20.8 
19.3 
17.9 
16.7 

14.9 

13.7 

12.2 

10.7 

9.3 

7.8 

6.4 
4.9 
3.4 
2.0 



Diff. 



0^ 



Diff. 



9 34. 
9 34. 
9 35. 
9 37. 
9 41. 
9 46. 

9 53. 

10 1. 
10 11. 
10 22. 
10 35. 

10 49. 

11 5. 

11 22. 

11 41. 

12 1. 

12 22. 

12 45. 

13 9. 

13 35. 

14 2. 

14 31, 

15 1, 

15 32 

16 5, 

16 39, 

17 14, 

17 51. 

18 29, 

19 9. 
19 50, 

0° 



0.5 
1.0 
2.4 
3.9 

5.4 

6.9 

8.3 

9.8 

11.2 

12.7 

14.3 

15.6 
17.1 
IS. 6 
20.0 

21.4 

22.9 
24.3 
25.7 
27.2 

28.6 
29.9 
31.4 
32.7 
34.1 

35.5 

36.8 
38.2 
39.5 
40.9 



19 50. 

20 32. 

21 15. 

22 0. 

22 46. 

23 33. 

24 22. 

25 12. 

26 3. 

26 55. 

27 48. 

28 43. 

29 38. 

30 35. 

31 33. 

32 32. 

33 32. 

34 33. 

35 35. 

36 39. 

37 43. 

38 48. 

39 54. 

41 1. 

42 9. 

43 18. 

44 28. 

45 39. 

46 51. 

48 3 

49 16 

0° 



42.1 
43.4 

44,8 
46.0 

1 47. 2 

|48.5 
149.8 
^50.9 

S52.2 
J53.3 

|54.5 

!55.6 

56.8 
58.0 
59.0 

60.0 

61.2 

62.2 
63.2 
64.3 

65.2 

66.2 
67.1 
68.1 
69.0 

69.8 
70.7 

71.6 
72.3 
73.2 



49 16. 

50 30. 
5145. 

53 0. 

54 16. 

55 33 

56 50. 

58 9. 

59 27^ 
047. 

2 6. 

3 27 

4 48. 

6 9 

7 31. 

8 53, 

10 15. 

11 3S. 

13 2. 

14 26. 

15 49. 

1714. 

18 38. 
20 3. 
2128, 
22 53, 

24 18 

25 43 

27 9 

28 34 
30 

1° 



73.9 

74.7 
75.5 
76.1 
76.8 

77.4 

78.1 
78.7 
79.3 
79.8 

80.3 
80.9 
81.3 
81.8 

82.2 

82.6 

83.0 

83.4 
83.7 
83.9 

84.3 

84.4 
84.7 
84.9 
185.1 

|85.1 

85.3 

85.4 
85.4 
85.5 



72 



TABLE LI. 

Equation of Moon's Centre. 
Argument. Anomaly corrected. 



Os 



lis 



III* 



I TVs 



V» 



forlO 10 " 



Diff 
forlO 



12 c 



Diff 

forlO 



13 ; 



Diff 
for 10 



Diff q o 
for 10 9 



Diff 

for 10 



0. 
30 3 32. 

10 7 5 
30 10 37, 

2 1410 
30 1 7 42, 

3 21 15 
30 24 17. 

4 28 19 
30 3151. 

5 0,35 23 

30 38 54 

6 42 25 
30 45 56 

7 49 27. 
30 5258. 

S 5628. 

30 59 58. 



> 



120 57.9 
J23 55.6 

26 52.2 
29 47.7 
32 42.0 

35 35.2 





3 ,v 

n-70. 



38 27. 
4118. 
44 7. 
46 56. 



170.5 



70. 
70. 

70. 
70. 



o 

6 

49 43.2 



52 29 
55 13 
57 57 



9 3 28 
30 6 57 

10 10 26 

304354 

11 1722 

30 20 50 

12 24 17 
30 12744. 



13 3110.2 
30 34 35. S 

14 38 1.0 
30 4125.6 

15 0]4449.6 



I 



9 
jby.7 
169.6 

69.5 

|69.3 
69.2 
69. 
68. 

65. 

6S. 
68. 
68. 
68. 



39 
3 20. 

5 59. 

8 37. 
11 14. 
13 50. 
16 24. 

18 57. 
21 28. 
12358. 
126 27. 
28 54. 
| 

,3120. 
33 44. 
36 7. 
'38 29. 
.'40 49. 



59.2 
58.9 
5S.5 
58 J 
57.7 
57.3 

57.0 
56.5 

56.1 
55.7 

55.3 

54.9 
54.5 
,54.0 

3 53.6 

53.2 



52.7 
52.3 
51.8 
51.4 

50.9 

50.5 
50.0 
49.6 
49.1 

48.6 

48.1 

47.7 
47.2 
46.6 



38 43. 
40 14. 
4142. 
,43 9. 
'4434. 
45 58. 

47 20. 

48 40. 

49 58. 
5115. 

52 30. 

53 43. 

54 54. 

56 *4. 

57 12. 
53 IS. 

59 22. 
25. 
126. 

2 25. 

3 23. 

41S. 

5 12. 

6 4. 
6 54. 
743. 

8 30. 

9 15. 

9 58. 

10 40. 

11 19, 



J 30.1 
" 29.6 
' 29.0 

9 2S - 4 
. 27.8 



27.3 



26.7 
2 26.1 
' 25.5 
t 25.0 

' 24.4 

3 

, 23.8 

' 23.2 

i. h'.6 

. 22.1 
o 

121.5 
9 
« 20.9 

5 20.3 
°.|19.7 

18.6 

I|l7.9 

1 17.4 
9 16.S 

5J 16 - 2 
|l5.6 

1 150 
* 14.4 

6 13.8 
i 13.3 



17 35.2 
17 20.9 
17 4.8 
1647.1 
16 27.6 
16 6.5 

15 43.7 
15 19.2 

14 53.1 
11425.2 
13 55.8 



?^|35.2 

i? nl 36 - 
11 0.4o fi4 

91L1 369 
7 20.5 db - 9 
37.3 
5 28.7 
335.6 6lt 

141.3 3S1 
38.5 



13 24.7 Q 

i25i.9 !;■ 

12 17.4 .. 

H41.4 ;■ 

11 3.7 p- 

13. 

10 24 3 nQ 

943.4 }J 
9 O.SIJJ 

8 16.6 ;*■ 

730.Sj • 

15. 

6 43.4 lfi 

5 54.4 J® 

5 3.9 ; • 
4H.7 ;• 

3 18.0i 

18. 

2 22.7 . 

125.5 19 - 



27.4 



59 27.4 
58 25.9 



59 45.8 
57 49.1 

5551.1 

53 52.0 
51 51.7 
149 50.3 
J4747.6 

45 43.8 
43 3S.9 
;4132.S 
39 25.6 
37 17.3 



35 7.9 
32 57.4 
30 45.8 
28 33.1 
26 19.4 

24 4.6 
214S.8 

19 31.9 
17 14.1 
14 55.2 



38.9 
39.3 

39.7 
40.1 
;40.5 
40.9 

141.3 

41.7 | 

42.0 
42.4 

42. S 
43.1 

43.5 

43.9 
'44.2 
44.6 

44.9 

45.3 
45.6 
45.9 
46.3 



5S2&.9 
55 43.8 
52 58.0 
50 11.6 
47 24.5 
44 36.8 

,41 4S.5 
38 59.5 
36 10.0 
33 19.8 
30 29.1 

J27 37.8 

24 45.9 



55.0 
55.3 
55.5 
55.7 
55.9 

56.1 

56.3 
56.5 
56.7 
56.9 

57.1 

57.3 



2153.5?:f 
19 0.6*™ 

16 7.1 ; '- 8 

5S.0 



13 13.1 

10 18.6 

7 23.6 

4 2S.1 

1 32.2 



58 35.8 
55 38.9 

5241.7 
49 43.9 
46 45.8 

43 47.3 
40 48.4 
37 49.1 
,3449.5 
3149 4 



5S.2 
58.3 
58. 
58.6 

5S.S 

59.0 
59.1 
59.3 
59.4 

59.5 

59.6 
59.8 
59.9 
60.0 



11 : 



13 : 



12 : 



IV 



TABLE LI. 

Equation of Moon'' s Centre. 
Argument. Anomaly corrected. 



73 



Vis 



VII* 



VIII* 



IX* 



XI* 



Diff 

for 10 



o 


30 

1 
30 

2 
30 

3 
30 

4 
30 

5 

30 

6 
30 

7 
30 

8 
30 

9 
30 

10 



-Mei.8 

546 61.8 
49.2 



30 55 
11 0J52 
JO 49 



12 
30 

13 
30 

14 
30 

15 



43.9 
38.6 
33.4 

28.1 
23.0 
18.0 
13.0 
8.1 

3.4 
58.8 
54.3 
50.0 

45.8 

41.9 
38.0 
34.4 
31.0 

27.8 

24.9 
22.2 
19.7 
17.5 
15.6 

14.0 
12.6 
11.6 
10.9 
10.6 



61.8 
61.8 
61.7 

61.8 

61,7 
61.7 
61.7 
61.6 
61.6 

61.5 
61.5 
61.4 
61.4 
61.3 

61.3 

61.2 
61.1 
61.1 

61.0 

60.9 
60.8 
60.7 
60.6 

60.5 

60.5 
60.3 
60.2 
60.1 



Diff 
forlO 



Diff 
for 10 



Diff 

for 10 



Diff 
for 10 



Diff 

for 10 



131.1 



58 46.7 
5Q 3.0 
53 20.0 
50 37.7 
47 56.2 

4515.4 
42 35.3 
39 56.0 
3717.4 
34 39.6 

32 2.7 
29 26.5 
2651.1 
2416.6 
2142.9 

19 10.0 
16 38.0 
14 6.9 
1136.6 
9 7.3 

6 38.9 
411.3 

144.7, 



59 18.9 
56 54.2 

5430.4 

52 7.5 
49 45.6 
47 24.7 
45 4.8 



54.8 
54.6 
54.3 
54.1 
53.8 
53.6 

53.4 
53.1 
52.9 
52.6 

52.3 

52.1 
51.8 
51.5 
51.2 
51.0 

50.7 
50.4 
50.1 
49.8 
49.5 

49.2 
48.9 
48.6 
48.2 
47.9 

47.6 
47.3 
47.0 
46.6 



43 39.2 
41 55.0 
40 12.0 
38 30.5 
36 50.3 
3511.3 

33 33.7 

31 57.5 
30 22.6 
28 49.0 
2716.8 

25 46.1 
2416.7 

22 48.7 
2122.1 
19 56.9 

18 33.1 
1710.8 
1549.8 
1430.4 
1312.5 

1155.9 

10 40.9 

9 27.3 

8 15.2 

7 4.6 

5 55.4 
447.8 
341.7 
2 37.1 
134.1 

1° 



34.7 
34.3 
33.8 
33.4 
33.0 
32.5 

32.1 
31.6 
31.2 
30.7 
30.2 

29.8 
29.3 

28.9 

28.4 

27.9 

27.4 
27.0 
26.5 
26.0 
25.5 

25.0 
24.5 
24.0 
23.5 

23.1 

22.5 
22.0 
21.5 
21.0 



42 24.8 
4212.1 
42 1.2 
41 52.0 
41 44.4 
4138.7 

41 34.6 
41 32.2 
4131.6 
41 32.7 

41 35.6 

4140.1 
4146.4 
4154 

42 4.3 
4215.9 

42 29.2 

42 44.2 

43 1.1 
4319.6 

43 39.9 

44 2.0 
4425.9 

44 51.5 
4518.8 

45 48.0 

46 18.9 

46 51.5 

47 26.0 

48 2.2 
48 40.1 



I0 C 



4.2 
3.6 
3.1 
2.5 
1.9 

1.4 

0.8 
0.2 
0.4 

1.0 

1.5 

2.1 
2.7 
3.3 
3.9 

4.4 

5.0 
5.6 

6.2 
6.8 
7.4 

8.0 
8.5 
9.1 
9.7 
10.3 

10.9 
11.5 
12.1 
12.6 



2116.4 
22 48.5 
2422.2 
25 57.7 
27 34.8 

29 13.7 

30 54.2 
32 36. 
34 20.2 

36 5.6 

37 52.8 

39 41.5 
41 32.0 
43 24.0 
4517.7 
4712.9 

49 
51 8.3 
53 8.4 
55 10.1 
5713.3 

591SJ2 
"124.5 
3 32.4 
541.9 
7 52.9 

10 5.5 
1219.5 
1435.1 
1652.1 
19 10.7 



30.7 
31.2 
31.8 
32.4 
33.0 
33.5 

34.0 
34.6 
35.1 
35.7 

36.2 

36.8 
37.3 
37.9 
38.4 
39.0 

39.5 
40.0 
40.6 
41.1 

41.6 

42.1 
42.6 
43.2 
43.7 

44.2 

44.7 
45.2 
45.7 
46.2 



2.1 

0.S 
0.7 
1.7 
3.7 
6.7 



57_10 
015.8 
321.8 
6 28.8 
9 36.8 

12 45.7 
15 55.5 
19 6.2 
2217 
25 30.3 

28 43.7 
3157.8 
3512.9 
38 28.7 
41 45.2 

45 2.6 
48 20.7 
5139.6 
54 59.1 
5819.3 



140.3 

5 1.9 

8 24.1 

1146.9 

1510.4 



59.6 
60.0 
60.< 
60.7 
61.0 
61.3 

61.7 
62.0 
62. 
62.7 

63.0 

63.3 
63.6 
63.9 
64.2 

64,5 

64.7 
65.0 
65.3 
65.5 

65.8 

66.0 
3 

66.5 
66.7 

67.0 

67.2 

67.4 
67.6 
67. 



74 



TABLE LI. 
Equation of Moon's Centre. 
Argument. Anomaly corrected. 



O 



IP 



III« 



IVs 



Ys 



Diff 
for 10 



ll c 



Diff 
forlO 



13° 



15 44 49.6 
30 48 13.1 

16 5135.9 



30154 58 
17 0|58 19 
30T40 



18 
30 

19 
30 

20 

30 

21 
30 

22 
30 

23 

30 

24 
30 

25 

30 

26 
30 

27 
30 

28 
30 

29 
30 



5 0. 

8 20. 
1139. 
14 57. 
18 14. 

2131. 
2447. 
28 2. 
31 16. 
34 29. 

37 42. 
40 53. 
44 4, 
4714. 
50 23. 

53 31. 

56 38. 
59 44. 



2 49 
5 53 

8 56 
1158 
14 59 
1759 



30 ( 20 57.9 
'10° 



67.8 
67.6 
67.4 
67.2 
67.0 
66.7 

66.5 
66.3 
66.0 
65.8 

65.5 

65.3 
65.0 
64.7 
64.5 

64.2 

63.9 
63.6 
63.2 
630 

62.7 

62.3 

62.0 
61.7 
61.3 

61.0 

60.7 
60.3 
60.0 
59.6 



4049.3 
43 7.9 
45 24.9 
47 40.5 
49 54.5 
52 7.1 

5418.1 
56 27.6 
58 35.5 



041.8 
2 46.7 

449.9 

651.6 

8 51.7 

10 50.2 

12 47.1 

1442.3 
16 36.0 

18 28.0 
20 18.5 

22 7.2 

23 54.4 
25 39 
27 23.7 

29 5.8 

30 46.3 

32 25.2 

34 2.3 

35 37.8 
3711.5 
38 43.6 

12° 



46.2 
45.7 
45.2 
44.7 
44.2 

43.7 

43.2 

42.6 
42.1 
41.6 
41.1 

40.6 
40.0 
39.5 
39.0 

38.4 

37.9 
37.3 

36.8 
36.2 

35.7 

35.1 
34.6 
34.1 
33.5 

33.0 

32.4 
31.8 
31.2 
30.7 



Diff 
forlO 



12 c 



Diff 
for 10 



ll c 



Diff 
forlO 



Diff 
for 10 



11 19.9 
1157.8 

12 34.0 

13 8.5 
1341.1 
1412.0 

1441.2 
15 8.5 
15 34.1 

15 58.0 

16 20.1 

16 40.4 

16 58.9 
1715.8 
1730.8 
1744.1 

17 55.7 

18 5.5 
18 13.6 
18 19.9 
18 24.4 

18 27.3 
18 28.4 
18 27.8 
18 25.4 
1821.3 

1815.6 
18 8.0 

17 58.8 
17 47.9 
17 35.2 

13° 



12.6 
12.1 
11.5 
10.9 
10.3 

9.7 

9.1 

8.5 
8.0 
7.4 

6.8 

6.2 
5.6 
5.0 

4.4 

3.9 

3.3 

2.7 
2.1 
1.5 
1.0 

0.4 
0.2 
0.8 
1.4 
1.9 

2.5 
3.1 
3.6 
4.2 



58 25.9 
57 22.9 
5618.3 
55 12.2 
54 4.6 
52 55.4 

51 44.8 
50 32.7 
4919.1 

48 4.1 
46 47.5 

45 29.6 
4410.2 
42 49.2 
4126.9 
40 3.1 

38 37.9 
3711.3 
35 43.3 
34 13.9 
32 43.2 

31 11.0 
29 37.4 
28 2.5 
26 26.3 

2448.7 

23 9.7 
2129.5 
19 48.0 
18 5.0 
16 20.8 

12° 



21.0 
21.5 
22.0 
22.5 
23.1 
23.5 

24.0 
24.5 
25.0 
25.5 

26.0 

26.5 
27.0 
27.4 
27.9 

28.4 

28.9 
29.3 
29.8 
30.2 
30.7 

31.2 
31.6 
32.1 
32.5 

33.0 

33.4 
33.8 
34.3 

34.7 



1455.2 

12 35.3 

1014,4 

752.5 

5 29.6 

3 5.8 

041.1 



58 15.3 
55 48.7 
5321.1 
50 52.7 

48 23.4 
45 53.1 
43 22.0 
40 50.0 
38 17.1 

35 43.4 
33 8.9 
30 33.5 

27 57.3 
25 20.4 

22 42.6 
20 4.0 
17 24.7 
14 44.7 
12 3.8 

9 22.3 
6 40.0 
3 57.0 
113.3 



58 28.9 



46.6 
47.0 
47.3 

47.7 
47.9 

48.3 

48.6 
48.9 
49.2 
49.5 

49.8 

50.1 
50.4 
50.7 
51.0 
51.2 

51.5 
51.8 
52.1 
52.3 

52.6 

52.9 
53.1 
3.3 
53.6 
53.8 

54.1 
54.3 
54.6 
54.8 



3149.4 
28 49.1 
25 48.4 
22 47.4 
19 46.0 
16 44.4 

13 42.5 
10 40.3 

7 37.8 
435.1 
132.2 



58 29.0 
55 25.6 
52 22.0 
49 18.1 
46 14.2 

43 10.0 
40 5.7 
37 1.2 
33 56.6 
30 51.9 

2747.0 
2442.0 
21 37.0 
1831.8 
15 26.6 

12 21.4 
9 16.1 
610.8 
3 5.4 
0.0 



60.1 
60.2 
60.3 
60.5 
60.5 

60.6 

60.7 
60.8 
60.9 
61.0 
61.1 

61.1 
61.2 
61.3 
61.3 

61.4 

61.4 
61.5 
61.5 
61.6 

61.6 



61.7 

61.8 
61.8 
61.8 
61.8 



TABLE LI. 

Equation of Moon's Centre. 
Argument. Anomaly corrected. 



75 



VI* 



VII* 



VIII* 



IX* 



X* 



XIs 



Diff 
for 10 



Diff: 
forlO, 



Diff 
forlO 



Diff 

for 10 



2° 



Diff 
for 10 



Diff 
for 10 



15 0,28 10.6 
302510.5 



30 19 11.6 

17 1612.7 
30 1314.2 

18 0:1016.1' 
30 718.3 

19 0; 421.1 
30 124.2 

20 5327~8 



60.0 
59.9 

159.8 
59.6 
59.5 

1 59.4 

59.3 
59.1 
59.0 

58.8 
58.6 



53. 5 



30 55 31.9 
51 0'52 36.4 

30 49 41.4 ™'t 
22 46 46.9 

30 43 52.9 



23 40 59.4 
3033 6.5 

24 3514.1 
30 32 22.2 

25 29 30.9 i 



30 

26 

30 



26 40.2 
23 50.0 
21 0.5 



27 0jl8 11.5| 
30,15 23.2 



28 
30 

29 
30 

30 



12 35.5 
9 48.4 
7 2.0 
416.2 
131.1 



58.2 
58.0 

57.8 

57.6 
57.5 
57.3 

57.1 

56.9 

56.7 
56.5 
56.3 
56.1 

55.9 

55.7 
55.5 
55.3 
55.0 



45 4.8 
42 45.9 
40 28.1 
3811.2 
35 55.4 
33 40.6 

3126.9 
29 14.2 
27 2.6 
24 52.1 

22 42.7 

20 34.4 

18 27.2 
1621.1 
1416.2 
1212.4 

10 9.7 
8 8.3 
6 8.0 
4 8.9 
210.9 

014.2 



58 18.7 
50 24.4 
3431.3 
52 39.5 

50 4S.9 
48 59.6 
4711.5 
45 24.7 
43 39.2 

1° 



46.3 
45.9 
45.6 
45.3 
44.9 
44.6 

44.2 
43.9 
43.5 
43.1 
42.8 

42.4 
42.0 
41.6 
41.3 
40.9 

40.5 
40.1 
39.7 
39.3 

38.9 

38.5 
38.1 
37.7 
37.3 
36.9 

36.4 
36.0 | 
35.6| 
35.2 1 



134.1 
32.6 



59 32.61 
58 34.2 
57 37.3 
56 42.o! 

5548.3 

54 56.1 1 
54 5.6 
5316.6 

52 29.2 



51 43.4 
50 59.2 
5016.6 
49 35.7 
4S56.3 

48 18.6 
47 42.6 
47 8.1 
46 35.3 
46 4.2 

45 34.8 
45 6.9 
4440.8 
4416.3 
43 53.5 

43 32.4 
4312.9 
42 55.2 
42 39.1 

42 24.8 



20.5 
20.0 
19.5 
19.0 
18.4 
17.9 

17.4 
16.8 
16.3 
15.S 

15.3 

14.7 
14.2 
13.6 
13.1 

12.6 

12.0 
11.5 
10.9 
10.4 

9.8 

9.3 

8.7 
8.21 
7.6 

7.0 

6.5 
5.9 

5.4 
4.8 



48 40.1 ]C 

49 19.9 J^ 

14 4 

50 44.6 

5129.7 
52 16.5 



15.0 
15.6 

16.2 

16.8 
17.4 
17.9 
18.6 

19.1 

19.7 
20.3 

20.9 
21.5 

22.1 

22.6 
23.2 
23.8 
24.4 

25.0 

25.5 
26.1 
26.7 
27.3 

27.8 

L4 29'0 
13 17.3 ^ 

19 46.0^- \ 

21 16.4 6UA 



53 5.1 
53 55.4 
5447.5 

55 41.3 

56 37.0 

57 34.3 

58 33.5 

1)37.1 

141.5 

2 47.7 

3 55.6 

5 5.3 

6 16.7 

7 29.8 

8 44.7 
10 J. 3 
1119.71 

12 39.8 

14 1.6 

15 25.1 



1910.7 
2130.6 
23 52.1 
2615.1 
28 39.5 
31 5.3 



46.6 

47.2 
47.7 
48.1 
48.6 
49.1 



33 32.5 
36 1.2 



49.6 
3S31.2J*™ 

43 35.5 ° U ' y 
51.4 
46 9.7 
48 45.2 
5122.1 
54 0.3 
56 39.9 

59 20.7 



2 2.8 

446.2 

730.9 

1016.8 

13 4.0 
15 52.4 
1842.0; 
2132.9 
24 24.8 

J2718.0] 

30 12.3! 



7.8j 
4.4! 
2.l| 



51.S 
52.3 
52.7 
53.2 

53.6 

54.0 
54.5 
54.9 
55.3 
55.7 

56.1 
56.5 
57.0 

157.3 

157.7 

58.1 
58.5 
58.9 
59.2 



15 10.4 
8 34.4 
2159.0 
25 24.2 
28 49.8 
32 16.0 

35 42.7 
39 9.9 
42 37.5 
46 5.5 
49 34.0 

53 2.8 
56 31.9 



1.6 
3 31.5 
7 1.8 

10 32.3 
14 3.1 
17 34.2 
21 5.5 
2437.0 

23 ! 
3140.7 
35 12.8 
38 45.1 
4217.3 

45 49.7 

49 22.2 
52 54.S 

50 27 A 
0.0 



68.0 
68.2 
68.4 
68.5 
68.7 
68.9 

69.1 

69.2 
69.3 
69.5 

69.6 

69.7 

69.9 
70.0 
70.1 

70.2 

70.3 
70.4 
70.4 
70.5 

70.6 

70.6 
70.7 
70.7 
70.8 
70.8 

70.8 
70.9 
70.9 
70.9 



76 



TABLE LII. 

Variation. 



Argument. Variation, corrected. 



O 



U* 



Ills 



IVs 



V* 



DifF. 1 : 



DifF. 1° 



33 

1 39 
2 ! 40 
3J41 

4 ! 42 



5 44 



10 

11 
12 
13 
14 
15 

16 
1? 
IS 
19 
20 

21 
22 
23 
24 
25 



0.0 
13.3 
26.5 
39.5 
52.2 

4.5 

16 

27.7 
38.4 
43.3 
57.4 

5.6 
12.8 
18.9 
23.8 
27.5 

29.8 
30.7 
30.1 
28.0 
24.2 



1S.7 
11.4 
2.3 
51.2 
38.2 



26 5 

27 6 

28 6 

30 1 8 



23.1 
6.01 

46.7 

29, 7 25.2 

1.5 



73.3 
73.3 

73.0 
72.7 
72.3 

71.9 

71.3 
70.7 
69.9 
69.1 
6S.S 

67!2 
66.1 
64.9 
63.7 

62.3 

60.9 
59.4 
57.9 

56.2 

54.5 

52.7 
50.9 
48.9 
47.0 
44.9 

42.9 
40.7 
38.5 
36.3 



8 1.5 

8 35.5 

9 7.2 
9 36.5 

10 3.4 
10 27.9 

10 49.9 
111 9.4 

11 26.4 
11 40.9 

11 52.9 

12 2.2 
12 9.0 
12 13.2 
12 14. 
12 13.9 

12 10.3 
12 4.2 
11 55.5 
11 44.2 
11 30.5 

11 14.1 
10 55.3 
10 34.0 
10 10.2 
9 44.0 

9 15.4 
8 44.5 

5 11.2 
7 35.7 

6 57.9i 



34.0 
J31.7 
29.3 
1 26.9 
24.5 

22.0 

19.5 
17.0 

14.5 
12.0 

9.3 

6.8 
4.2 
1.6 
0.9 
3.6 

6.1 

8.7 

11.3 

13.7 

16.4 

18.8 
21.3 
23.8 
26.2 

28.6 

30.9 
33.3 
35.5 
37.8 



! Dlff. 0: 
I 



DifF. 



DifT. 



6 57.9 
6 18.0 
5 35.9- 

I 4 51.7! 

, 4 5.5 
3 17.3 

2 27.2 1 

1 35.3^ 

l 41.6 

59 46.1! 

,58 49.0; 

[57 50.21 
156 50.0; 
,55 48.3 
54 45.2 
|53 40.9 



39.9 
42.1 
44.2 
46.2 
48.2 

50.1 

51.9 
53.7 
55.5 
57.1 

58.8 

60.2 
61.7 
163.1 
!64.3 

J65.6 
|52 35.3J firs 
51 2S.5 ™° 
50 20.7 HI 
49 11 9 y 
48 2.21 



135 54.4 
34 40.4 
33 26.6 
32 13.0 
30 59.6 
29 46.7 

28 34.3 
27 22.4 
J26 11.2 
25 0.7 
23 51.1 



J |70.5 

46 51.7 
45 40.5' \i 
44 28.6 '^ 
:43 16.1 III 
42 3.2 7 - J 
73.3 
'40 49.9' 7 
|39 36.2 £' 
38 22.4 ^« 

i 37 8 - 4 740 
35 54.41 ' 4>U 

0° 



22 42.3 
21 34.5 
20 27.9 
19 22.3 
,18 1S.0 

17 15.0 
16 13.4 
15 13.2 
14 14.6 

13 17.5J 

12 22.2' 

11 2S.5I 

10 3G.7 

9 46.8 

8 58.8 

S 12.7 
1 7 28.7 
I 6 46.8 
! 6 7.1 
I 5 29.5 

loo 



74.0 
73.8 
73.6 
1 73.4 
72.9 
72.4 

71.9 
71.2 
70.5 
69.6 

68.8 

67.8 
66.6 
65.6 
64.3 
63.0 

61.6 

60.2 
5S.6 
57.1 

55.3 

53.7 
51.8 
49.9 
48.0 

46.1 

44.0 
41.9 
39 7 
37.6 



5 29.5 
4 54.2 
421.3 
3 50.6 
3 22.3 
2 56.5 

2 33.1 

2 12.1 
153.7 
137.8 
1 1 24.5 

1 13.7 
1 5.5 
1 0.0 
57.0 
^56.7 

59.0 
il 3.9 

1 11.5 
1 21.6 
134.4 



35.3 

32.9 
30.7 

28.3 
25.8 
23.4 

21.0 
18.4 
15.9 
13.3 

10.8 

8.2 
5.5 
3.0 
0.3 

2.3 

4.9 

7.6 

10.1 

12.8 

15.4 



18.0 
20.5 
23.1 
25.5 



'1 49.8 
2 7.8 
;2 2S.3 
2 51.4 
i3 16.9 

J28.1 

4 48.5j 354 

5 23.9 q~ ~ 
16 l.6 6t 



6 1.6 
641.6 

7 23.9 

8 8.4 

8 55.0 

9 43.7 



Diff. 



40.0 
42.3 
44.5 
46.6 
48.7 

50.8 

1222.0?;' 

13 18.6|^ 

14 16.9 ° 8 - 3 

60.1 



15 17.0 

16 18.7 

17 22.0 

18 26.9 

19 33.1 

20 40.7 

21 49.6 

22 59.6 

24 10.8 

25 22.9 

|26 35.9 
27 49.8 

29 4.5 

30 19.7 

31 35.6 



32 51.9 

34 8.6 

35 25.6 

36 42.7 
38 0.0 



61.7 
63.3 
64.9 
66.2 

67.6 

68.9/ 

70.0 

71.2 

72.1 

73.0 

73.9 

74.71 
75.2; 
75.9 1 

76.3 ' 

76.7 
77.0 
77.1 



TABLE LIL 

Variation. 



77 



Argument. Variation corrected. 



Vis 



VII* 



VIII« 



IX* 



X* 



XI« 



Diff. 1° 



Diff. 1° 



Diff. Oo 



Diff. 



Diff. 



Diff 



33 

39 17 

40 34 
4151 

43 8 

44 24 

45 40 

46 55 

48 10 

49 24 

50 37 

5149 
53 
5410 

55 19 

56 26 

57 33 

58 38 

59 41 



43 
143 

2 41 
S38 
432 
5 25 
616 

7 5 

7 51 

8 36 

9 18 
9 58 



77.3 
77.1 
77.0 
76.7 
76.3 

75.9 

75.2 
74.7 
73.9 
73.0 

72.1 

71.2 
70.0 
6S.9 
67.6 

66.2 

64.9 
63.3 
61.7 
60.1 
58.3 

56.6 
54.7 
52.8 
50.8 

48.7 

46.6 
44.5 
42.3 
40.0 



9 5S.4 

10 36.1 

11 11.5 
1144.4 

12 15.0 

12 43.1 

13 8.6 
1331.7 

13 52.2 
1410.2 
1425.6 

14 38.4 
1448.5 

14 56.1 

15 1.0 
15 3.3 

15 3.0 
15 0.0 
14 54.5 
1446.3 
14 35.5 

1422.2 
14 6.3 
13 47.9 
13 26.9 
13 3.5 

12 37.7 
12 9.4 
1138.7 
11 5.8 
10 30.5 

1° 



37.7 
35.4 
32.9 
30.6 
28.1 

25.5 

23.1 
20.5 
1S.0 
15.4 

12.8 



10 30.5 
9 52.9 
913.2 
S31.3 
747.3 
7 1.2 

6 13.2 
5 23.3 
431.5 
3 37.8 
242.5 



145.4 
46.S 



10.1 

7.6 

49 |5y4b.b 

23 |58 45.0 



0.3 I 

3.0 

5.5 

8.2 
10.8 

13.3 

15.9 

18.4 
21.0 
23.4 

25.8 

28.3 
30.7 
32.9 
35.3 



'57 42.0 

156 37.7 

155 32.1 
54 25.5 
5317.7 

'52 8.9 

■50 59.3 
!49 48.8 
! 48 37.6 
47 25.7 
46 13.3 

i45 0.4 
43 47.0 
42 33.4 
4119.6 
40 5.6 



37.6 
39.7 
41.9 
44.0 
46.1 
48.0 

49.9 
51.8 
53.7 
55.3 

57.1 

58.6 
60.2 
61.6 
63.0 

64.3 

65.6 
66.6 
67.8 
68.8 
69.6 

70.5 
71.2 
71.9 
72.4 

72.9 

73.4 
J73.6 
73.8 
74.0 



40 5.6 
38 51.6 
37 37.6 
36 23.8 
3510.1 
33 56.8 

32 43.9 
3131.4 

30 19.5 
29 8.3 
27 57.8 

26 48.1 
25 39.3 |R , 
2431.51®!- 

23 24.7 J J" 

22 19.1 bb 

64. 

21 14.8 RQ 

20 11.7 °f 

1910.0!°;. 

is wBJ 

1711.0 bt 



74 
74 
73 
73 
73 

72 

72.5 
71.9 
71.2 
70.5 
69.7 
68.8 



9 2. 

3 24. 
748. 
715. 
6 44. 
6 16. 

5 49. 
5 26. 
5 4. 
445. 

4 29. 

415. 
4 4. 
3 55. 



0^ 



16 13.9 
15 18.4 
1424.7 
13 32.8 
12 42.7 

1154.5 
ill 8.3 

10 24.1 
9 42.0 
9 2.1 

0o 



6 
3 

1 
7 
2 
8 
57.1 



3 46.1 



3 45. 
3 46. 
351. 

3 57. 

4 7. 

419. 
4 33. 

4 50. 

5 10. 
5 32. 

5 56. 

6 23. 

6 52. 

7 24. 

7 58. 



37.8 
35.5 
33.3 
30.9 
28.6 
26.2 

23.8 
21.3 
18.S 
16.4 

13.7 

11.3 
8.7 
6.1 
3.6 

0.9 

1.6 
4.2 
6.8 
9.3 

12.0 

14.5 

17.0 
19.5 
22.0 
24.5 

26.9 
29.3 
31.7 
34.0 



7 5S.5 

8 34.8 

9 13.3 
9 54.0 

10 36.9! 



1121.8 

12 8.8 

12 57.7 

13 48.6 
1441.3 

15 35.8 

16 32.0 



36.3 
33.5 
40.7 
42.9 
144.9 

47.0 

148.9 
150.9 
52.7 
54.5 

56.2 



57. 



17299 59.4 
19 30.2 6 °- 9 



20 32.5 

2136.2 
2241.1 
123 47.2 
'24 54.4 

26 2.6 

27 11.7 
23 21.6 

29 32.3 

30 43.6 

31 55.5 

33 7.8 

34 20.5 

35 33.5 

36 46.7 
38 0.0 

0° 



62.3 
63.7 

64.9 
66.1 
67.2 
68.2 
69.1 

69.9 

70.7 
71.3 
71.9 

72.3 

72.7 
73.0 
73.2 
73.3 



78 TABLE LIII. Reduction. 

Argument. Supplement of Node + Moon's Orbit Longitude. 



OsVIs Diff. Is VIIsDiff. IlsVIIIs Diff. IIIsIXs Diff. IVsXs Diff. Vs XIs Diff. 



0.0 
45.6 
31.2 
16.9 

2.6 
48.4 

34.3 

20.3 

6.4 

52.6 

39.0 

25.6 
12.3 
59.3 
46.5 
33.9 

21.6 
9.5 

57.7 
46.2 
35.0 

24.2 
13.7 
3.5 
53.7 
44.2 

35.2 

26.5 

18.3 

10.4 

3.0 



14.4 
14.4 
14.3 
14,3 
14.2 
14.1 

14.0 
13.9 
13.8 
13.6 

13.4 

13.3 
13.0 
12.8 
12.6 

12.3| 

12.1 
11.8 
11.5 
11.2 

10.8 

10.5 

10.2 

9.8 

9.5 

9.0 

8.7 
8.2 
7.9 
7.4 



3.0 

56.0 
49.5 
43.4 
37.8 
32.7 

28.2 
23.9 
20.0 
16.8 
14.1 

11.8 
10.1 



8.1 






8.8 





10.1 


o 


11.8, 





14.1 





16.8 





20.0 





23.9 





28.2 





32.7 


37.8 


43.4 





49.5 





56.0 


1 


3.0 



7.0 
6.5 
6.1 
5.G 
5.1 

4.5 

4.3 
3.9 
3.2 

2.7 

2.3 

1.7 
1.3 
0.7 
0.3 

,0.3 

0.7 
1.3 

1.7 

2.3 

2.7 

3.2 
3.9 
4.3 
4.5 

5.1 

5 6 

1 

6.5 
7.0 



3.0 

10.4 
18.3 
26.5 
35.2 
44.2 

53.7 
3.5 
13.7 
24.2 
35.0 

46.2 
57.7 
9.5 
21.6 
33.9 

46.5 
59.3 
12.3 
25.6 
39.0 

52.6 

6.4 

20.3 

34.3 

48.4 

2.6 
16.9 
31.2 
45.6 

0.0 



9 

9 

9 
10 
10 
10. 

11. 

11. 
11. 

12. 

12. 

12. 

12. 
13. 
13. 
13. 

13. 

13. 
13. 
14. 

14. 
14. 

14. 
14. 
44, 

14. 



0.0 

14.4 
28.8 
43.1 
57.4 
11.6 

25.7 
39.7 
53.6 
7.4 
21.0 

34.4 
47.7 
0.7 
13.5 
26.1 

38.4 
50.5 
2.3 
13.8 
25.0 

35.8 
46.3 
56.5 
6.3 
15.8 

24.8 
33.5 
41.7 
49.6 
57.0 



14.4 
14.4 
14.3 
14.3 
14.2 
14.1 

14.0 
13.9 
13.8 
13.6 
13.4 

13.3 
13.0 

12.8 
12.6 

12.3 

12.1 
11.8 
11.5 
11.2 
10.8 

10.5 

10.2 

9.8 

9.5 

9.0 

8.7 
8.2 
7.9 
7.4 



57.0 
4.0 
10.5 
16.6 
22.2 
27.3 

31.8 
36.1 
40.0 
43.2 
45.9 

48.2 
49.9 
51.2 
51.9 

52.2 

51.9 

51.2 
49.9 
48.2 
45.9 

43.2 
40.0 
36.1 
31.8 
27.3 

22.2 
16.6 
10.5 
4.0 
570 



12 57.0 

12 49.6 

12 41.7 

12 33.5 

12 24.8 

12 15.8 



6.3 
56.5 
46.3 
35.8 
25.0 

13.8 
2.3 
50.5 
38.4 
26.1 



10 13.5 

10 0.7 

9 47.7 

9 34.4 

9 21.0 



7.4 
53.6 
39.7 

25.7 
11.6 

57.4 
43.1 
28.8 
14.4 
0.0 



7.4 
7.9 
8.2 
8.7 
9.0 
9.5 

9.8 
10.2 
10.5 
10.8 
11.2 

11.5 

11.8 
12.1 
12.3 

12.6 

12.8 
13.0 
13.3 
13.4 
13.6 

13.8 
13.9 
14.0 
14.1 
14.2 

14.3 
14.3 
144 
.14 4 



TABLE LIV. Lunar Nutation in Longitude. 
Argument. Supplement of the Node. 





O 


Is 


lis 


Ills 


TVe 


Vs 






+ 


+ 


+ 


+ 


+ 


+ 




o 


„ 


,, 


,, 


f> 


'/ 


// 


o 





0.0 


8.5 


14.8 


17.3 


15.2 


8.8 


30 


2 


0.6 


9.0 


15.1 


17.2 


14.9 


8.1 


28 J 


4 


1.2 


9.4 


15.4 


17.2 


14.5 


7.7 


26 


6 


1.7 


10.0 


15.6 


17.2 


14.2 


7.2 


24 


8 


2.3 


10.4 


15.9 


17.2 


13.8 


6.5 


22 


10 


2.9 


10.9 


16.4 


17.1 


13.5 


6.1 


20 


12 


3.5 


11.4 


16.3 


17.0 


13.0 


5.4 


18 


14 


4.1 


11.8 


16.5 


16.9 


12.6 


4.8 


16 


16 


4.6 


12.2 


16.7 


16.7 


12.2 


4.3 


14 


18 


5.2 


12.6 


16.8 


16.5 


11.8 


3.7 


12 


20 


5.8 


13.1 


16.9 


16.4 


11.3 


3.0 


10 


22 


6.2 


13.4 


17.1 


16.2 


10.9 


2.4 


8 


24 


6.9 


13.8 


17.1 


15.9 


10.4 


1.8 


6 


26 


7.4 


14.1 


17.2 


15.7 


9.8 


1.3 


4 


28 


7.8 


14.5 


17.2 


15.4 


9.4 


0.6 


2 


30 


8.5 


14.8 


17.3 


15.2 


8.8 


0.0 







XIa 


Xs 


IX* 


VIII* 


VII* 


VI* 





TABLE LV. 



79 



Moon's Distance from the North Pole of the Ecliptic. 
Argument. * Supplement of Node+Moon's Orbit Longitude. 



Ill* 



IV* 



Vis 



YIls 



VIII* 



84 c 



S5 : 



Diff. 
for 10 



S7 C 



Diff. 

for 10 



Diff. 

for 10 



<j.>- 



IDiff. 

ifor 10 



94 c 




30 

1 
30 

2 
30 

3 

30 

4 
30 

5 

30 

6 
30 

7 



8 
30 

9 
30 

10 

30 

11 
30 

12 
30 

13 
30 

14 
30 

15 



3916.0 
39 16.7 
39 18.8 
39 22.4 
39 27.3 
39 33.7 

39 41.5 

39 50.6 

40 1.2 
40 13.2 
40 26.7 

40 41.5 

40 57.7 

41 15.4 
4134.4 

41 54.8 

42 16.7 

42 39.9 

43 4.6 
43 30.6 

43 58.1 

4426.9 

44 57.1 

45 28.S 

46 1.8 
46 36.1 

4711.9 
4749.0 

48 27.5 

49 7.4 

J49 48.7 

!84° 



20 42.7 

22 4.2 

23 27.0 

24 51.0 

26 16.2 

27 42.6 

29 10.1 

30 38.9 

32 8 

33 39.9 
3512.2 

36 45.6 

38 20.1 

39 55.8 
41 32.7 

43 10.6 

44 49.7 
46 29.9 

48 11.2 

49 53.5 
51 37.0 

53 21.6 

55 7 

56 53.8 
58 41.6 



30.3 

2 20.1 
4H.0 
6 2.9 
755.7 
9 49.6 



27.2 
27.6 
28.0 
28.4 
28.8 
29 2 

29.6 
30.0 
30.4 
30.8 
31.1 

31.5 
31.9 
32.3 
32.6 

33.0 

33.4 
33.8 
34.1 
34.5 
34.9 

35.2 
35.7 
35.9 
36.2 
36.6 

37.0 
37.3 
37.6 
38.0 



S6 : 



13 46.6 

16 6.9 
18 27.8 
20 49.5 
23 11.8 
25 34.8 

27 58.5 
30 22.8 
32 47.7 
35 13.2 
37 39.3 

40 6.1 
42 33.4 
45 1.2 
47 29.6 
49 58.6 

52 28.1 
54 58.2 
57 28.7 
59 59.8 
231.3 

5 3.3 

7 35.8 
10 8.8 
12 42.1 
15 16.0 

17 50.2 
20 24.9 
22 59.9 
25 35.3 

28 11.1 

88° 
O* 



46.8 
47.0 
47.2 
47.4 
47.7 

47.9 

48.1 
48.3 
48.5 
48.7 
48.9 

49.1 
49.3 
49.5 
49.7 

49.8 

50.0 
50.2 
50.4 
50.5 
50.7 

50.8 
51.0 
51.1 
51.3 

51.4 

51.6 
51.7 
51.8 
51.9 



48 0.0 
5041.4 
53 22.9 
56 4.3 
58 45.7 



127.0 

4 8.3 
6 49.5 
9 30.6 

1211 

14 52.5 

1733.3 

20 14.0 
22 54.4 
25 34 

28 14 

30 54.9 
33 34.7 
36 14.3 
38 53.7 
41 32.8 

4411.7 
46 50.4 
49 28.7 
52 6 
5444.6 

57 22.1 

59 59.3 

2 36.2 

512.7 

748.9 



53.8 
53.8 
53.8 
53.8 
53.8 
53.8 

53.7 
53.7 
53.7 
53.6 

53.6 

53.6 
53.5 
53.5 
53.4 

53.3 

53.3 
53.2 
53.1 
53.0 

53.0 

52.9 
52.8 
52.7 
52.6 

52.5 

52.4 
52.3 
52.2 
52.1 



2213.4 
24 33.1 
26 52.2 



4<\.6 
46.4 
46.0 



2910.2,,, 
3127.5|f 5 8 6 
3344.2 *° D 
45.3 

45.0 



36 0.2 
38 15.3 
40 29.7 
42 43.3 
44 56.2 

47 8.1 
49 19.4 
5129.7 
53 39.3 
55 48.0 

57 55.8 



91 ( 



XI* 



2.8 
2 8.9 
414.1 
618.4 

8 21.8 
10 24.3 
12 25.9 
1426.6 
16 26.3 

18 25.0 
20 22.8 
22 19.7 
2415.5 
26 10.4 

93° 

X* 



44.5 
44.3 
44.0 

43.8 
43.4 
43.2 
42.9 
42.6 

42.3 
42.0 
41.7 
41.5 
41.1 

40.8 
40.5 
40.2 
39.9 
39.6 

39.3 
38.0 

38.6 
38.3 



1517.3 

16 37.7 

17 56.8 

19 14.6 

20 31.3 
2146.7 

23 0.8 
2413.7 

25 25.3 

26 35.7 
2744.8 

28 52.6 

29 59.0 

31 4.3 

32 8.2 
3310.9 

3412.2 

35 12.2 

36 10 

37 8.3 

38 4.4 

38 59.1 

39 52.5 

40 44.6 

41 35.3 

42 24.7 

4312.7 

43 59.4 
4444.7 

45 28.7 

46 11.3 



30 
30 

29 
30 

28 
30 

27 
30 

26 
30 

25 

30 
24 

30 
23 

30 

22 
30 

21 
30 

20 

30 
19 

30 
18 

30 

17 
30 

16 
30 

15 



94 c 



IX* 



80 TABLE LV. 

Moon's Distance from the North Pole of the Ecliptic* 
Argument. Supplement of Node -f- Moon's Orbit Longitude. 



I III* 


IV* 




V* 




VI* 




VII* 




VIII* 






84° 


86° 


Diff. 
for 10 


88° 


Diff. 
for 10 


91° 


Diff. 
for 10 


93° 


Diff. 
for 10 


94° 




O '1 , „ 


/ /, 




. ,, 




, „ 




/ „ 




, ,r 


' 


15 49 48.7 


9 49.6 


38.3 
38.6 
39.0 
39.3 
39.6 


28 11.1 


52.1 
52.2 
52.3 
52.4 
52.5 


7 48.9 


51.9 
51.8 
51.7 
51.6 
51.4 


26 10.4 


38.0 
37.6 
37.3 
37.0 

36.6 


46 11.3 


15 


30 50 31.3 ill 44.5 


30 47.3 


10 24.7 


28 4.3 


46 52.6 


30 


16 51 15.3-13 40.3 


33 23.8 


13 0.1 


29 57.1 


47 32.5 


14 


30 52 0.6 15 37 2 


36 0.7 


15 35.1 


31 49.0 


48 11.0 


30 


17 52 47.3 17 3o.0 


38 37.9 


18 9.8 


33 39.9 


48 48.1 


13 


30 53 35.3 \ 19 33.7 


41 15.4 


20 44.0 


35 29.7 


49 23.9 


30 


1 | 


39.9 




52.6 




51.3 




36.2 






18 54 24.7,21 33.4 


40.2 


43 53.2 


52.7 


23 17.9 


51.1 
51.0 
50.8 
50.7 


37 18.4 


35.9 


49 58.2 


12 


30 55 15.4J23 34.1 


40.5 
40.8 
41.1 


46 31.3 


52.8 
52.9 
53.0 


25 51.2 


39 6.2 


35.6 
35.2 
34.9 


50 31.2 


30 


19 56 7.5 25 35.7 


49 9.6 


28 24.2 


40 52.9 


51 2.9 


11 


30 57 0.9,27 38.2 


51 48.3 


30 56.7 


42 38.4 


51 33.1 


30 


20 57 55.6:29 41.6 


54 27.2 


33 28.7 


44 23.0 


52 1.9 


10 


1 | 


41.4 




53.0 




50.5 




34.5 






30 58 51.7 31 45.9 


41.7 
42.0 
42.3 
42.6 


57 6.3 


53.1 
53.2 
53.3 
53.3 


36 0.2 


50.4 
50.2 
50.0 
49.8 


46 6.5 


34.1 
33.8 
33.4 
33.0 


52 29.4 


30 


21 59 49.1!33 5i.! 


59 45.7 


38 31.3 


47 48.8 


52 55.4 


9 


30 47.8; 35 57.2 


2 25.3 


41 1.8 


49 30.1 


53 20.1 


30 


22 1 47.8 38 4.2 


5 5.1 


43 31.9 


51 10.3 


53 43.3 


8 


30 2 49.1 


40 12.0 




7 45.1 




46 1.4 


52 49.4 




54 5.2 


30 


1 




42.9 




53.4 




49.7 




32.6 






23 3 51,8 


42 20.7 


43.2 
43.4 
43.6 
44.0 


10 25.2 


53.5 
5-3.5 
53.6 
53.6 


48 30.4 


49.5 
49.3 
49.1 
48.9 


54 27.3 


32.3 
31.9 
31.5 
31.1 


54 25.6 


7 


30 4 55.7 


44 30.3 


13 5.6 


50 58.8 


56 4.2 


54 44.6 


30 


24 6 1.0 46 40.6 


15 46.0 


53 26.6 


57 39.9 


55 2.3 


6 


30 7 7.4 4S 51.9 


18 26.7 


55 53.9 


59 14.4 


55 18.5 


30 


25 8 15.2 51 3.8 




21 7.5 




58 20.7 




47 8 ™ ~ 


55 33.3 


5 


30 




44.3 

44.5 
44.8 
45.0 
45.3 


23 48.4 


53.6 

53.7 
53.7 
53.7 
53.7 




48.7 

48.5 
48.3 
48.2 
47.9 


u w -°|30.8 

2 20.l| 304 

3 51.2 5n a 


55 46.8 


30 


9 24.3 


53 16.7 


46.8 


26 


10 34.7 


55 30.3 


26 29.4 


3 12.3 


55 58.8 


4 


30 


11 46.3 


57 44.7 


29 10.5 


5 37.2 


5 21.1 


29.6 
29.2 


56 9.4 


30 


27 


12 59.2 


59 59.8 


3] 51.7 


8 1.5 


6 49.9 


56 18.5 


3 


30 


14 13.3 


2 15.8 


45.6 


34 33.0 


53.8 


10 25.2 


47.7 


8 17.4 


28.8 


56 26.3 


30 


28 


15 28.7 


4 32.5 


45.8 
46.0 
46.4 
46.6 


37 14.3 


53.8 

53.8 
53.8 


12 48.2 


47.4 
47.2 
47.0 
46.7 


9 43.8 


28.4 

28.0 


56 32.7 


2 


30 


16 45.4 


6 49.8 


39 55.7 


15 10.5 


11 9.0 


56 37.6 


30 


29 


18 3.2 


9 7.8 


42 37.1 


17 32.2 


12 33.0 


56 41.2 


1 


30 19 22.3 


11 26.9 


45 18.6 


19 53.1 


13 55.5*1" 
15 17.3f *'* 


56 43.3 


30 


30 20 42.7 

1 


13 46.6 


48 0.0 


538 22 13.4 


56 44.0 





'85° 


87° 




89° 


|92° 




94° j 


94° 




II* 


I* 




O 


I XI* 




X* 




IX* 





TABLE LVI. 

Equation II of the Moon's Polar Distance. 
Argument II, corrected. 



81 



III* diff. IV 



13.8 
13.9 
14.1 
14.5 
15.1 
15.8 

16.7 
17.7 
18.9 
20.3 
21.8 

23.5 
25.3 
27.3 
29.4 
31.7 

34.2 
36.8 
39.6 
42.5 
45.5 

48.7 
52.1 
55.6 

59.3 

1 3.1 

1 7.0 
1 11.1 

1 15.4 
1 19.8 
1 24.4 



0.1 
0.2 
0.4 
0.6 
0.7 

0.9 

1.0 
1.2 
1.4 
1.5 

1.7 

1.9 

2.0 
2.1 
2.3 

2.5 

2.6 
2.8 
2.9 
3.0 

3.2 

34 
3.5 
3.7 
3.8 
3.9 

4.1 
4.3 
4.4 
4.6 



II* 



diff. V* diff VI* diff. 



1 24.4 
1 29.0 
1 33.8 
1 38.7 
143.8 
149.0 

54.3 

1 59.8 

2 5.4 
2 11.1 
2 16.9 

2 22.9 
2 29.0 
2 35.2 
2 41.5 
2 47.9 

2 54.5 

3 1.1 
3 7.9 
3 14.8 
3 21.8 

3 28.8 
3 36.0 
3 43.3 
3 50.7 

3 5S.2 

4 5.8 
4 13.4 
4 21.2 
4 29.0 
4 36.9 

I* 



4.6 
4.9 

4.9 
5.1 
5.2 

5.3 

5.5 
5.6 

5.7 

5.S 

6.0 

6.1 

6.2 
6.3 
6.4 

6.6 

6.6 
6.S 
6.9 
7.0 

7.0 

7.2 
7.3 
7.4 
7.5 

7.6 

7.6 
7.8 
7.8 
7.9 



4 36.9 
4 44.9 

4 53 

5 1.1 
5 9.3 
5 17.6 

5 26.0 
5 34.4 
5 42.9 

5 51.4 

6 0.0 

6 8.7 
6 17.4 
6 26.2 
6 35.0 
6 43.8 

6 52.7 

7 1.6 
7 10.6 
7 19.6 
7 28.6 

7 37.7 
7 46.8 

7 55.9 

8 5.0 
8 14.1 

8 23.3 
8 32.5 
8 41.6 

8 50.8 

9 0.0 



S.9 
9.0 
9.0 
9.0 

9.1 

9.1 
9.1 
9.1 
9.1 

9.2 

9.2 

0.1 
9.2 
9.2 



o 



9 0.0 
9 9.2 

9 18.4 

9 27.5 
9 36.7 
9 45.9 

9 55.0 
10 4.1 
10 13.2 
10 22.3 
10 31.4 

10 40.4 

10 49.4 

10 58.4 

11 7.3 

11 16.2 

11 25.0 
11 33.8 
11 42 

11 51.3 

12 0.0 

12 8.6 
12 17.1 
12 25.6 
12 34.0 

12 42.4 

112 50.7 

112 5S.9 

113 7.0 
43 15.1 

13 23.1 



VII* 



9.2 
9.2 
9.1 
9.2 
9.2 
9.1 

9.1 
9.1 
9.1 
9.1 

9.0 

9.0 
9.0 

8.9 
S.9 

8,8 

8.8 
8.8 
8.7 
8.7 
8.6 

8 5 
9.5 

8.4 
8.4 



13 23.1 
13 31.0 
13 38.8 
13 46.6 

13 54.2 

14 1.8 



7.9 

7.S 
7.8 
7.6 
7.6 

7.5 
14 9.3 74 
14l6.7i.l-q 
14 24.0 11% 
14 31.2' 
14 38.2 



diff 



VIII* diff. 



| XI* 



14 45.2 
1,4 52.11 

14 58.91 

15 5.5 
15 12.1 

15 18.5 
15 24.8 
15 31.0 
15 37.1 
15 43.1 

15 48.9 

15 54.6 

16 0.2 1 
16 5.7 
16 11.0 

16 16.2 
16 21.3 
16 26.2 
16 31.0 
16 35.6 



7.0 

7.0 

6.9 
6.8 
6.6 
6.6 

6.4 

6.3 
6.2 
6.1 
6.0 

5.8 

5.7 
5.6 

5.5 
5 3 
5.2 

5.1 
4.9 
4.8 
4.6 



16 35.6 
16 40.2 
16 44.6 
16 48.9 
16 53.0 

16 56.9 

17 0.7 
17 4.4 
17 7.9 
17 11.3 
17 14.5 

17 17.5 
17 20.4 
17 23.2 
17 25.8 
17 2S.3 

17 30.6 
17 32.7 
17 34.7 
17 36.5 
17 38.2 

17 39.7 

17 41.1 
17 42.3 
i 17 43.3 
17 44.2 

S 17 44.9 
117 45.5 
17 45.9 
17 46.1 
17 46.2 



3.7 

3,5 
3.4 
3.2 

3 

2.9 

2.8 

2. 

2.5 

2.3 

2.1 
2.0 
1.8 

1.7 

1.5 

1.4 
1.2 
1.0 
0.9 
0.7 

0.6 
0.4 
0.2 
0.1 



o 
30 

29 

28 
27 
26 

25 

24 
23 

22 
21 
20 

19 
18 
17 
16 
15 

14 
13 
12 

11 

10 



I IX* 



TABLE LVII. 

Equation III of Moon's Polar Distance. 

Argument. Moon's True Longitude. 





III* 


IV* 


V* 


VI* 


VII* 


VIII* 




o 



16.0 


14.9 


L2.0 


8.0 


4.0 


1.1 


o 
30 


6 


16.0 


14.5 


11.3 


7.2 


3.3 


0.7 


24 


12 


15.8 


13.9 


10.5 


6.3 


2.6 


0.4 


18 


19 


15.6 


13.4 


9.7 


5.5 


2.1 


0.2 


12 


24 


15.3 


12.7 


8.8 


4.7 


1.5 


0.0 


6 


30 


14.9 


12.0 


8.0 


4.0 


1.1 


0.0 







II* 


I* 


0* 


XI* 


X* 


IX* 





82 TABLE LVIII. 

To convert Degrees 
and Minutes into 
Decimal Parts. 



TABLE LIX. 

Equations of Moon's Polar Distance. 
Arguments, Arg. 20 of Long. ; V to IX 

corrected; X not corrected; and XI 

and XII corrected. 



Beg. 


Dec 


&Mm. 


parts. 


o f 




1 5 


003 


1 26 


4 


1 48 


5 


2 10 


6 


2 31 


7 


2 53 


8 


3 14 


9 


3 36 


10 


3 58 


11 


4 iy 


12 


4 41 


13 


5 2 


14 


5 24 


15 


5 46 


16 


6 7 


17 


6 29 


18 


6 50 


19 


7 12 


20 


7 34 


21 


7 55 


22 


8 17 


23 


8 38 


24 


9 


25 


9 22 


26 


9 43 


27 


10 5 


28 


10 26 


29 


10 43 


30 


11 10 


31 


1131 


32 


1153 


33 


12 14 


34 


12 36 


35 


12 58 


36 


13 19 


37 


13 41 


38 


14 2 


39 


14 24 


40 


14 46 


41 


15 7 


42 


15 29 


43 


15 50 


44 


16 12 


45 


16 34 


46 


16 55 


47 


17 17 


48 


17 38 


49 


18 


50 


18 22 


51 


18 43 


52 


19 5 


53 



Arg 


20 


V 


VI 


VII 


vm 


IX 


X 


XI 


Arg 


Arg 


XII 


-j 

Arg. 










,, 





— 


„ 


„ 


„ 







// 




250 


0.3 


55.9 


6.1 


2.6 


25.1 


3.0 


0.7 


0.9 


250 





4.0 


500 


260 


0.3 


55.8 


6.2 


2.7 


25.1 


3.1 


0.7 


0.9 


240 


10 


3.7 


510 


270 


0.4 


55.7 


6.3 


2.8 


25.0 


3.2 


0.8 


1.0 


230 


20 


3.4 


520 


280 


0.6 


55.4 


6.5 


3.0 


24.9 


3.5 


1.0 


1.0 


220 


30 


3.1 


530 


290 


0.8 


55.1 


6.9 


3.3 


24.8 


3.8 


1.2 


1.1 


210 


40 


2.8 


540 


300 


1.0 


54.6 


7.3 


3.7 


24.7 


4.3 


1.5 


1.2 


200 


50 


2.5 


550 


310 


1.3 


54.1 


7.8 


4.2 


24.4 


4.9 


1.8 


1.3 


190 


60 


2.3 


560 


320 


1.7 


53.4 


8.4 


4.7 


24.1 


5.6 


2.2 


1.4 


180 


70 


2.1 


570 


330 


2.1 


52.7 


9.1 


5.4 


23.8 


6.4 


2.7 


1.5 


170 


80 


1.9 


580 


340 


2.6 


51.9 


9.8 


6.1 


23.5 


7.2 


3.2 


1.7 


160 


90 


1.7 


590 


350 


3.1 


51.0 


10.7 


6.9 


23.2 


8.2 


3.8 


1.9 


150 


100 


1.6 


600 


360 


3.7 


50.0 


11.6 


7.7 


22.8 


9.2 


4.4 


2.1 


140 


110 


1.5 


610 


370 


4.3 


48.9 


12.6 


8.7 


22.4 


10.3 


5.1 


2.3 


130 


120 


1.5 


620 


380 


4.9 


47.7 


13.6 


9.7 


21.9 


11.5 


5.8 


2.5 


120 


130 


1.5 


630 


390 


5.6 


46.5 


14.8 


10.7 


21.4 


12.8 


6.6 


2.8 


110 


140 


1.5 


640 


400 


6.4 


45.2 


16.0 


11.8 


20.9 


14.1 


7.4 


3.0 


100 


150 


1.6 


650 


410 


7.1 


43.9 


17.2 


13.0 


20.4 


15.5 


8.3 


3.3 


90 


160 


1.7 


660 


420 


7.9 


42.5 


18.5 


14.2 


19.9 


17.0 


9.1 


3.5 


80 


170 


1.9 


670 


430 


8.8 


41.0 


19.8 


15.5 


19.3 


18.5 


10.1 


3.8 


70 


180 


2.1 


680 


440 


9.6 


39.5 


21.2 


16.8 


18.7 


20.1 


11.0 


4.1 


60 


190 


2.3 


690 


450 


10.5 


38.0 


22.6 


18.1 


18.1 


21.7 


12.0 


4.4 


50 


200 


2.5 


700 


460 


11.3 


36.4 


24.1 


19.4 


17.5 


23.3 


12.9 


4.7 


40 


210 


2.8 


710 


470 


12.2 


34.9 


25.5 


20.8 


16.9 


24.9 


13.9 


5.0 


30 


220 


3.1 


720 


480 


13.2 


33.2 


27.0 


22.2 


16.3 


26.6 


15.0 


5.4 


20 


230 


3.4 


730 


490 


14.1 


31.6 


28.5 


23.645.6 


28.3 


16.0 


5.7 


10 


240 


3.7 


740 


500 


150 


30.0 


30.0 


25 15.0 


30.0 


17.0 


6.0 





250 


4.0 


750 


510 


15 9 


28.4 


31.5 


26.4 14.4 


31.7 


18.0 


6.3 


990 


260 


4.3 


760 


520 


16.8 


26.8 


33.0 


27.8,13.7 


33.4 


19.0 


6.6 


980 


270 


4.6 


770 


530 


17.8 


25.1 


34.5 


29.2 13.1 


35.1 


20.1 


7.0 


970 


280 


4.9 


780 


540 


18.7 


23.6 


35.9 


30.6^2.5 


36.7 


21.1 


7.3 


960 


290 


5.2 


790 


550 


19.5 


22.0 


37.4 


31.9 


11.9 


38.3 


22.0 


7.6 


950 


300 


5.5 


800 


560 


20.4 


20.5 


38.8 


33.2 


11.3 


39.9 


23.0 


7.9 


940 


310 


5.7 


810 


570 


21.2 


19.0 


40.2 


34.5 


10.7 


41.5 


23.9 


8.2 


930 


320 


5.9 


820 


580 


22.1 


17.5 


41.5 


35.8 


10.1 


43.0 


24.9 


8.5 


920 


330 


6.1 


830 


590 


22.9 


16.1 


42.8 


37.0 


9.6 


44.5 


25.7 


8.7 


910 


340 


6.3 


840 


600 


23.6 


14.8 


44.0 


38.2 


9.1 


45.9 


26.6 


9.0 


900 


350 


6.4 


850 


610 


24.4 


13.5 


45.2 


39.3 


8.6 


47.2 


27.4 


9.2 


890 


360 


6.5 


860 


620 


25.1 


12.3 


46.4 


40.3 


8.1 


48.5 


28.2 


9.5 


880 


370 


6.5 


870 


630 


25.7 


11.1 


47.4 


41.3 


7.6 


49.7 


28.9 


9.7 


870 


380 


6.5 


880 


640 


26.3 


10.0 


48.4 


42.3 


7.2 


50.8 


29.6 


9.9 


860 


390 


6.5 


890 


650 


26.9 


9.0 


49.3 


43.1 


6.8 


51.8 


30.2 


10.1 


850 


400 


6.4 


900 


660 


27.4 


8.1 


50.2 


43.9 


6.5 


52.8 


30.8 


10.3 


840 


410 


6.3 


910 


670 


27.9 


7.3 


50.9 


44.6 


6.2 


53.6 


31.3 


10.5 


830 


420 


6.1 


920 


680 


28.3 


6.6 


51.6 


45.3 


5.9 


54.4 


31.8 


10.6 


820 


430 


5.9 


930 


690 


28.7 


5.9 


52.2 


45.8 


5.6 


55.1 


32.2 


10.7 


810 


440 


5.7 


940 


700 


29.0 


5.4 


52.7 


46.3 


5.3 


55.7 


32.5 


10.8 


800 


450 


5.5 


950 


710 


29.2 


4.9 


53.1 


46.7 


5.2 


56.2 


32.8 


10.9 


790 


460 


5.2 


960 


720 


29.4 


4.6 


53.5 


47.0 5.1 


56.5 


33.0 


11.0 780 


470 


4.9 


970 


730 


29.6 


4.3 


53.7,47.2 5.0 


56.833.2 


11.0 770 


480 


4.6 


980 


740 


29.7 


4.2 


53.8,47.3, 4.9 


56.9,33.3 


11.1 


760 


490 


4.3 


990 


750 


29.7 


4.1 


53.9|47.4 4.9 


57.0'33.3 


11.1 


750 


500 


4.0 


1000 



Constant 10" 



TABLE LX. TABLE LXI. 83 

Small Equations of Moon's Parallax. Moon's Equatorial Parallax. 

Args., 1, 2, 4, 5, 6, 8, 9, 12, 13, of Long. Argument. Arg. of Evection. 



A.) 1 

00.0 
3 0.0 
6J0.0 

90.1 

12 0.1 
150.1 

18 0.2 
21 0.3 
24'0.4 

! 
27 0.5 
30 0.5 
33 0.6 

1 
36 0.7 
39 0.7 
42 0.8 

45 0.8 
48 0.8 
50 0.S 


2 

1.6 
1.6 
1.5 

1.5 
1.4 
1.3 

1.1 
1.0 
0.9 

0.7 
0.6 
0.4 

0.3 
0.2 
0.1 

0.0 
0.0 
0.0 


4 

0.6 
0.6 
0.6 

0.6 
0.5 
0.5 

0.4 
0.4 
0.3 

0.3 
0.2 
0.2 

0.1 
0.1 
0.0 

0.0 
0.0 
0.0 


5 

1.6 
1.6 
1.5 

1.5 
1.4 
1.3 

1.1 
1.0 
0.9 

0.7 
0.6 
0.4 

0, 

0.0 
0.0 
0.0 


6 

1.9 
1.9 

1.8 

1.8 

1.7 
1.6 

1.4 
1.3 
1.2 

1.0 
0.9 
0.7 

0.6 
0.5 
0.4 

0.3 

0.3 
0.3 


8 

0.0 
0.0 
0.0 

0.1 
0.2 
0.2 

0.3 
0.5 
0.6 

0.7 

0.8 
0.9 

1.0 
1.1 
1.1 

1.2 
1.2 
1.2 


9 

rr 

3.6 
3.5 
3.1 

2.6 
1.9 
1.3 

0.7 
0.2 
0.0 

0.1 
0.4 
0.8 

1.5 
2.1 

2.8 

3.2 
3.5 
3.6 


12 

1.4 
1.4 

1.4 

1.3 
1.2 

1.1 

1.0 
0.9 
0.7 

0.6 
0.5 
0.4 

0.3 

0.2 
0*1 

0.0 
0.0 
0.0 


13 

2.0 

2.0 
1.9 

1.8 
1.7 
1.6 

1.4 
1.2 
1.0 

0.9 
0.7 
0.5 

0.4 
0.2 
0.1 

0.0 
0.0 
0.0 


A. 

100 
97 
94 

91 

88 
85 

82 
79 
76 

73 
70 
67 

64 
61 

5S 

55 
52 
50 


Constant 7" 

The first two figures only of the Arguments 
are taken. 

_ _. . 





O 


Is 


II* 


Ills 


IV* 


V* 








, „ 


, ,, 


/ // 


" 





1 20.8 


1 15.6 


1 1.5 


42.6 


24.1 


10.8 30 


1 


1 20.8 


1 15.2 


1 0.9 


41.9 


23.6 


10.5 


29 


2 


1 20.8 


1 14.9 


1 0.3 


41.3 


23.0 


10.2 


28 


3 


1 20.7 


1 14.5 


59.7 


40.6 


22.5 


9.9 


2"! 


4 


1 20.7 


1 14.2 


59.2 


40.0 


21.9 


9.6 


26 


5 


1 20.6 


1 13.8 


58.6 


39.4 


21.4 


9.4 


25 


6 


1 20.6 


1 13.4 


57.9 


38.7 


20.9 


9.1 


24 


7 


1 20.51 13.0 


57.3 


38.1 


20.4 


8.8 


23) 


S 


1 20.4 1 12.6 


56.7 


37.4 


19.9 


8.6 


22, 


9 


1 20.3 1 12.2 


56.1 


36.8 


19.4 


8.4 


21j 


10 


1 20.2 1 11.7 

1 


55.5 


36.1 


18.9 


8.2 


20. 


11 


120.1 1 11.3 


54.9 


35.5 


18.4 


8.0 


19 


12 


1 19.9 1 10.8 


54.2 


34.9 


17.9 


7.8 


IS 


13 


1 19.8 1 10.4 


53.6 


34.2 


17.5 


7.6 


17 


14 


1 19.6 1 9.9 


53.0 


33.6 


17.0 


7.4 


16 


15 


1 19.5 1 9.4 


52.3 


33.0 


16.6 


7.2 


15 


16 


1 19.3 1 9.0 


51.7 


32.4 


16.1 


7.1 


14 


17 


1 19.1 1 8.5 


51.1 


31.7 


15.7 


6.9 


13 


IS 


1 18.9 1 8.0 


50.4 


31.1 


15.2 


6.S 


12 


19 


1 18.7 1 7.5 


49.8 


30.5 


14.8 


6.7 


11 


20 


1 18.4 1 7.0 


49.1 


29.9 


14.4 


6.5 


10 


21 


1 18.2 1 6.5 


48.5 


29.3 


14.0 


6.4 


9 


22 


1 18.0 1 5.9 


47.8 


28.7 


13.6 


6.3 


S 


23 


1 17.7 1 5.4 


47.2 


28.1 


13.2 


6.3 


7 


24 


1 17.41 4.8 


46.5 


27.5 


12.9 


6.2 


6 


25 


1 17.1 


1 4.3 


45.9 


26.9 


12.5 


6.1 


5 


26 


1 16.9 


1 3.8 


45.2 


26.3 


12.1 


6.1 


4 


27 


1 16.6 


1 3.2 


44.6 


25.8 


11.8 


6.1 


3 


28 


1 16.2 


1 2.6 


43.9 


25.2 


11.5 


6.0 


2 


29 


1 15.91 1 2.1 


43.3 


24.7 


11.1 


6.0 


1 


30 1 15.6|1 1.5 


42.6124.1 


10.8 


6.o o; 




XI* 


Xs 


IX* 


vniJvuJvi* 


j 



34 



TABLE LXII. 

Moon's Equatorial Parallax, 



Argument. Anomaly. 



diflf 



I* 



diff II* 



diff 



Ills 



diff 



IV* 



diff V* diff 



58 57.7 
58 57.7 
58 57.6 
58 57.4 
58 57.1 
58 56.8 

58 56.4 
58 56.0 
58 55.4 
58 54.8 
58 54.2 

58 53.4 
58 52.6 
58 51.8 
58 50.8 
58 49.8 

58 4S.7 
58 47.6 
58 46.4 
58 45.1 
58 43.8 

58 42.4 
58 40.9 
53 39.4 
58 37.8 
58 36.2 

58 34.4 
58 32.7 
58 30.9 
5S29.0 
58 27.0 

XI* 



58 27.0! 
58 25.0l; 
58 23.0: 
58 20.9 
5818.7 
5816.5 



58 14.3 
58 12.0 
58 9.6 
58 7.2 
5S 



4.8 



2.3 

2.4 

2.4 
2.4 

2.5 



58 2.3 
57 59.8* 
57 57.2* 
57 54.6; 
5751.9 ^ 
2 

2 



57 49.2 
57 46.4 
57 43.7 
57 40.8 
57 33.0 

5735 ll 
57 32.2i 

57 29.3J 
57 26.3| 
57 23.3 



2 
2 

|2.9 



7 57 20.2L 
' J5717.2 J 

9 5714.1^ 

5711.0^ 

57 7.9 6 

"1 x- r 



57 7.9 
57 4.8 
57 1.6 

56 58.4 
56 55.2 
56 52.0 

56 48.8 
56 45.5 
56 42.3 
56 39.0 
56 35.7 

56 32.4 
56 29.1 
56 25.8 
56 22.5 
56 19.2 

56 15.9 

5612.6 
56 9 
56 6.0 
56 2.7 

55 59.3 
55 56.0 
55 52.7 
55 49.4 
55 46.1 

5542.8 
55 39.6 
55 36.4 
55 33.1 
55 29.8 

IX* 



3.1 
3.2 
3.2 
3.2 
3.2 

3.2 

3.3 

3.2 
3.3 
3.3 

3.3 

3.3 
3.3 
3.3 
3.3 
3.3 

3.3 
3.3 
3.3 
3.3 

3.4 

3.3 
3.3 
3.3 
3.3 

3.3 

3.2 
3.2 
3.3 
3.3 



55 29.8 
55 26.6 
55 23.4 
55 20.2 
5517.0 
5513.8 

5510.6 
55 7.5 
55 4.4 
55 1.3 
54 58.2 

5455.1 
54 52.1 
5449.1 
5446.1 
5443.1 

5440.2 
54 37.3 
54 34.4 
5431.5 
54 28.7 

5425.9 
54 23.1 
54 20.3 
5417.6 
5414.9 

5412.2 
54 9.6 
54 7.0 
54 4.4 
54 1.9 

VIII* 



54 1. 
53 59. 
53 56. 
53 54. 
53 52. 
53 49. 

53 47. 
53 45. 
53 42. 

53 40. 
53 38. 

53 36. 
53 34. 
53 32. 
53 30. 
53 28. 

53 26. 
53 24 
53 22. 
53 20 
53 18. 

5317. 
5315 
53 13 
53 12. 
53 10 



2.5 

2.5 

2.4 
2.4 
2.4 
2.3 

2.3 
2.2 
2.3 

2.1 

2.2 

2.1 

2.1 
2.0 
2.0 

1.9 

1.9 
1.9 

1.8 
1.8 
1,8 

1.7 
1.6 
1.7 
1.6 

1.5 1 



53 3. 
53 1. 
53 0. 
52 59. 
52 58. 
52 57. 

52 55. 
52 54. 
52 53. 
52 52. 
52 51. 

52 51. 
52 50. 
52 49. 
52 48. 
52 47. 

52 47. 
52 46. 
52 46. 
52 45. 
52 45. 



*1.4 

5 



53 8.9 
53 7.4 



13 



VII* 



52 44.6 
152 44.2 
152 43.8 
5243.5 
52 43.3 

5243.1 
52 42.9 
52 42.8 
52 42.7 
,52 42.7 



VI* 



1.3 

1.2 
1.2 
1.1 

1.2 

ll.O 
1.0 
1.0 
0.9 

0.9 

0.9 
O.S 
0.7 
0.7 
0.7 

0.6 
0.6 
0.5 
0.5 

0.4 

0.4 

0.4 
0.3 
0,2 

0.2 

0.2 
0.1 
0.1 
0.0 



TABLE LXIII. 



85 



Moon's Equatorial Parallax. 
Argument. Argument of the Variation. 





O 


I* 


11^ 


III* 


IV* 


V* 




o 


// 


// 


„ 


// 


" 


r, 


o 





55.6 


42.3 


16.0 


3.7 


17.6 


44.0 


30 


1 


55.6 


41.5 


15.3 


3.8 


18.5 


44.8 


29 


2 


55.5 


40.7 


14.5 


3.8 


19.3 


45.6 


28 


3 


55.5 


39.8 


13.8 


3.9 


20.1 


46.3 


27 


4 


55.3 


39.0 


13.1 


4.1 


21.0 


47.0 


26 


5 


55.2 


38.1 


12.4 


4.3 


21.9 


47.7 


25 


6 


55.0 


37.2 


11.7 


4.5 


22.7 


48.4 


24 


7 


54.8 


36.3 


11.1 


4.7 


23.6 


49.1 


23 


8 


54.6 


35.5 


10.4 


5.0 


24.5 


49.7 


22 


9 


54.3 


34.6 


9.8 


5.3 


25.4 


50.3 


21 


10 


54.0 


33.7 


9.2 


5.6 


26.3 


50.9 


20 


11 


53.7 


32.7 


8.7 


6.0 


27.2 


51.5 


19 


12 


53.3 


31.8 


8.2 


6.3 


28.2 


52.1 


18 


13 


52.9 


30.9 


7.7 


6.8 


29.1 


52.6 


17 


14 


52.5 


30.0 


7.2 


7.2 


30.0 


53.1 


16 


15 


52.0 


29.1 


6.7 


7.7 


30.9 


53.5 


15 


16 


51.5 


28.2 


6.3 


8.2 


31.8 


54.0 


14 


17 


51.0 


27.2 


5.9 


8.7 


32.8 


54.4 


13 


18 


50.5 


26.3 


5.6 


9.3 


33.7 


54.8 


12 


19 


49.9 


25.4 


5.3 


9.8 


34.6 


55.1 


11 


20 


49.4 


24.5 


5.0 


10.5 


35.5 


55.4 


10 


21 


48.8 


23.6 


4.7 


11.1 


36.4 


55.7 


9 


22 


48.1 


22.7 


4.5 


11.7 


37.3 


56.0 


8 


23 


47.4 


21.9 


4.3 


12.4 


38.2 


56.2 


7 


24 


46.8 


21.0 


4.1 


13.1 


39.0 


56.4 


6 


25 


46.1 


20.1 


3.9 


13.8 


39.9 


56.6 


5 


26 


45.4 


19.3 


3.8 


14.5 


40.8 


56.8 


4 


27 


44.6 


18.5 


3.7 


15.3 


41.6 


56.9 


3 


28 


43.9 


17.6 


3.7 


16.1 


42.4 


56.9 


2 


29 


43.1 


16.8 


3.7 


16.8 


43.2 


57.0 


1 


30 


42.3 


16.0 


3.7 


17.6 


44.0 


57.0 







XI* 


X* 


IX* 


VIII* 


VII* 


VI* 





86 



TABLE LXIV. 



TABLE LXV. 



Reduction of the Parallax, 
and also of the Latitude. 

Argument. Latitude. 



Moon's Semi-diameter. 



Argument. 



Equatorial Parallax. 



Lat, 



Red. 
of par 



0.0 
0.0 

0.1 
0.3 
0.5 
0.7 

1.0 
1.4 
1.8 
2.3 

2.7 

3.3 
3.8 

4.4 
4.9 
5.5 

6.1 
6.7 

7.2 
7.8 
8.3 

8.8 

9.2 

9.7 

10.0 

10.3 

10.6 

10.8 
11.0 
11.1 
11.1 



Red. of 
Lat. 



0.0 

1 11.8 

2 22.7 

3 32.1 

4 39.3 

5 43.4 

6 43.7 

7 39.7 

8 30.7 

9 16.1 
9 55.4 

10 2S.3 

10 54.3 

11 13.2 
11 24.7 

11 28.7 

11 25.2 
11 14.1 
10 55.7 
10 30.0 
9 57.4 

9 18.3 
8 32.9 
7 42.0 
6 45.9 
5 45.4 

4 41.0 
3 33.5 
2 23.7 
1 12.3 
0.0 



Subsidiary Table. 



Lat. 


+ 3' 


o 


„ 





+ 0.0 


6 


0.0 


12 


0.0 


15 


0.0 


18 


0.1 


24 


0.1 


30 


0.1 


36 


0.2 


42 


0.2 


48 


0.3 


54 


0.3 


60 


0.4 


72 


0.5 


78 


0.6 


84 


0.6 


90 


+ 0.6 



3' 



0.0 
0.0 
0.0 
0.0 
0.1 
0.1 

0.1 
0.2 
0.2 
0.3 
0.3 

0.4 
0.5 
0.6 
0.6 
0.6 



Eq.Par Semidia. 


Eq.ParjSemidia. 


Eq.Par 


Semidia. 


sec 

1 


Pro. 
Par. 

0.3 


53 J 14 26.5 


56 


15 


15.6 


59 


16 4.6 


53 10 


14 29.3 


56 10 


15 


18.3 


59 10 


16 7.4 


2 


0.5 


53 20 


14 32.0 


56 20 


15 


21.0 


59 20 


16 10.1 


3 


0.8 


53 30 


14 34.7 


56 30 


15 


23.8 


59 30 


16 12.8 


4 


1.1 


53 40 


14 37.4 


56 40 


15 


26.5 


59 40 


16 15.6 


5 


1.4 


53 50 


14 40.2 


56 50 


15 


29.2 


59 50 


16 18.3 


6 


1.6 


54 


14 42.9 


57 


15 


31.9 


60 


16 21.0 


7 


1.9 


54 10 


14 45.6 


57 10 


15 


34.7 


60 10 


16 23.7 


8 


2.2 


54 20 


14 48.3 


57 20 


15 


37.4 


60 20 


16 26.4 


9 


2.4 


54 30 


14 51.1 


57 30 


15 


40.1 


GO 30 


16 29.2 


10 


2.7 


54 40 


14 53.8 


57 40 


15 


42.8 


60 40 


16 31.9 






54 50 


14 56.5 


57 50 


15 


45.6 


60 50 


16 34.6 






55 


14 59.2 


58 


15 


48.3 


61 


16 37.3 






55 10 


15 2.0 


5S 10 


15 


51.0 


61 10 


16 40.1 






55 20 


15 4.7 


58 20 


15 


53.7 


61 20 


16 42.8 






55 30 


15 7.4 


58 30 


15 


56.5 


61 30 


16 45.5 






55 40 


15 10.1 


58 40 


15 


59.2 


61 40 


16 48.2 






55 50 


15 12.9 


58 50 


16 


1.9 


61 50 


16 51.0 






56 


15 15.6 


59 


16 


4.6 


62 


16 53.7 







TABLE LXVI. 

Augmentation of Moon's Semi-diameter. 





Horizon. Semi-diameter. | 


Horizon 


. Semi-diameter. 


Alt 










'Alt. 




















14'30" 


15' 


16' 


17 i 


14' 30" 


15' 


16' 


17 


o 


,, 


„ 


,, 


" 1 ° 


// 


// 


// 


2 


0.6 


0.6 


0.7 


0.8 1 42 


9.2 


9.8 


11.2 


12.6 


4 


1.0 


1.1 


1.3 


1.5 J 45 


9.7 


10.4 


11.8 


13.3 


6 


1.5 


1.6 


1.9 


2.1 48 


10.2 


10.9 


12.4 


14.0 


8 


2.0 


2.1 


2.4 


2.7 I 51 


10.6 


11.4 


13.0 


14.7 


10 


2.4 


2.6 


3.0 


3.4 [54 


11.1 


11.8 


13.5 


15.2 


12 


2.9 


3.1 


3.6 


4.0 


;57 


11.5 


12.3 


14.0 


15.8 


14 


3.4 


3.6 


4.1 


4.7 


GO 


11 8 


12.7 


14.4 


16.3 


16 


3.8 


4.1 


4.7 


5.3 


63 


12.2 


13.0 


14.9 


16.8 


18 


4.3 


4.6 


5.2 


5.9 


66 


12.5 


13.4 


15.2 


17.2 


21 


4.9 


5.3 


6.0 


6.8 


69 


12.8 


13.7 


15.6 


17.6 


24 


5.6 


6.0 


6.8 


7.7 


72 


13.0 


13.9 


15.9 


17.9 


27 


6.2 


6.7 


7.6 


8.6 


75 


13.2 


14.1 


16.1 


18.2 


30 


6.9 


7.3 


8.4 


9.5 


78 


13.4 


14.3 


16.3 


18.4 


33 


7.5 


8.0 


9.1 


10.3 


81 


13.5 


14.4 


16.5 


18.6 


36 


8.1 


8.6 


9.8 


11.1 


84 


13.6 


14.5 


16.6 


18.7 


39 


8.6 


9.2 


10.5 


11.9 | 


90 


13.7 


14.6 


16.7 


18.8 



TABLE LXVII. 



87 



Moon's Horary Motion in Longitude. 
Arguments. 1 to 18 of Longitude. 



Arg. 


2 


3 


±_ 


5 


6 


_L_ 


7 


8 


9 


Arg. 




„ 


» 




,, 


,/ 




~~7, 


„ 


/, 







5.0 


0.0 


2.9 


1.9 


0.0 


0.00 


0.00 


0.00 


0.16 


100 


2 


5.0 


0.0 


2.8 


1.9 


0.0 


0.00 


0.00 


0.00 


0.15 


98 


4 


4.9 


0.0 


2.8 


1.9 


0.0 


0.01 


0.00 


0.02 


0.15 


96 


6 


4.8 


0.1 


2.8 


1.9 


0.1 


0.03 


0.01 


0.05 


0.14 


94 


a 


4.7 


0.2 


2.7 


1.8 


0.1 


0.06 


0.01 


0.09 


0.12 


92 


10 


4.5 


0.3 


2.6 


1.7 


0.2 


0.09 


0.02 


0.14 


0.10 


90 


12 


4.3 


0.4 


2.5 


1.7 


0.2 


0.13 


0.02 


0.19 


0.09 


88 


14 


4.1 


0.6 


2 3 


1.6 


0.3 


0.18 


0.03 


0.26 


0.07 


86 


16 


3.8 


0.7 


2.2 


1.5 


0.4 


0.23 


0.04 


0.33 


0.05 


84 


IS 


3.6 


0.9 


2.0 


1.4 


0.5 


0.28 


0.05 


0.41 


0.03 


82 


20 


3.3 


1.1 


1.9 


1.3 


0.6 


0.34 


0.06 


0.50 


0.02 


80 


22 


3.0 


1.3 


1.7 


1.1 


0.7 


0.40 


0.07 


0.58 


0.01 


78 


24 


2.7 


1.5 


1.5 


1.0 


O.S 


0.46 


0.08 


0.67 


0.00 


76 


26 


2.3 


1.7 


1.3 


0.9 


0.9 


0.52 


0.10 


0.77 


0.00 


74 


28 


2.0 


1.9 


1.2 


0.8 


1.0 


0.58 


0.11 


0.86 


0.00 


72 


30 


1.7 


2.1 


1.0 


0.7 


1.1 


0.63 


0.12 


0.94 


0.01 


70 


32 


1.4 


2.2 


0.8 


0.5 


1.2 


0.69 


0.13 


1.03 


0.01 


68 


34 


1.2 


2.4 


0.7 


0.4 


1.3 


0.74 


0.14 


1.11 


0.03 


66 


36 


0.9 


2.6 


0.5 


0.3 


1.3 


0.7S 


0.15 


1.18 


0.05 


64 


38 


0.7 


2.7 


0.4 


0.3 


1.4 


0.82 


0.16 


1.25 


0.06 


62 


40 


0.5 


2.8 


0.3 


0.2 


1.5 


0.86 


0.16 


1.30 


0.08 


60 


42 


0.3 


2.9 


0.2 


0.1 


1.5 


0.89 


0.17 


1.35 


0.10 


58 


44 


0.2 


3.0 


0.1 


0.1 


1.6 


0.91 


0.17 


1.39 


0.11 


56 


46 


0.1 


3.1 


0.0 


0.0 


1.6 


0.93 


0.18 


1.42 


0.12 


54 


48 


0.0 


3.1 


0.0 


0.0 


1.6 


0.94 


0.18 


1.44 


0.13 


52 


50 


0.0 


3.1 


0.0 


0.0 


1.6 


0.94 


0.18 


1.44 


0.13 


50 




Arg. 


10 


11 


12 


13 


14 


15 


16 


17 


i— 


[ Arg. 




., 


,, 


// 


„ 


,, 


// 


„ 


~~7, 


" 







0.00 


0.26 


0.00 


0.00 


0.00 0.00 


0.26 


0.00 


0.21 


I 100 


2 


0.00 


0.25 


0.00 


0.00 


0.00 0.00 


0.26 


0.00 


0.20 


1 98 


4 


0.02 


0.24 


0.01 


0.00 


0.01 iO.OO 


0.26 


0.00 


0.20 


| 96 


6 


0.04 


0.22 


0.03 


0.01 


0.C2 10.01 


0.25 


0.00 


0.20 


94 


8 


0.08 


0.20 


0.04 


0.02 


] 0.04' 0.01 


0.25 


0.01 


0.20 


92 


10 


0.12 


0.17 


0.07 


0.03 


0.06 


0.02 


0.24 


0.01 


0.20 


90 


12 


0.16 


0.14 


0.09 


0.04 


0.09 


0.02 


0.22 


0.02 


0.19 


88 


14 


0.20 


0.11 


0.12 


0.06 


0.12 


0.03 


0.21 


0.02 


0.19 


86 


16 


0.24 


0.03 


0.16 


0.07 


0.15 


0.04 


0.20 


0.03 


0.13 


84 


18 


0.28 


0.05 


0.19 


0.09 


1 0.19 


0.05 


0.19 


0.04 


0.13 


82 


20 


0.31 


i0.03 


0.23 


0.11 


0.22 


0.06 


0.17 


0.05 


0.17 


80 


22 


0.34 


0.01 


0.27 


0.13 


0.26 


0.07 


0.15 


0.06 


0.17 


78 


24 


0.35 


0.00 


0.31 


0.15 


0.30 


0.08 


0.14 


0.07 


0.16 


76 


26 


0.36 


jo.oo 


035 


0.17 


0.34 


0.08 


0.12 


0.07 


0.16 


74 


28 


0.35 


0.01 


0.39 


0.19 


0.38 


0.09 


0.11 


0.08 


0.15 


72 


30 


0.34 


0.02 


0.43 


0.21 


0.42 


0.10 


0.09 


0.09 


0.15 


70 


32 


0.32 


0.04 


0.47 


0.23 


0.45 


0.11 


0.07 


0.10 


0.14 


68 


34 


0.29 


0.06 


0.50 


0.25 


0.49 


0.12 


0.06 


0.11 


0.14 


66 


36 


0.26 


0.09 


0.54 


0.26 


0.52 


0.13 


0.05 


0.12 


0.13 


64 


38 


0.22 


0.11 


0.57 


0.28 


0.55 


0.14 


0.04 


0.12 


0.13 


62 


40 


0.18 


0.14 


0.59 


0.29 


0.58 


0.14 


0.02 


0.13 


0.12 


60 


42 


0.15 


0.16 


0.62 


0.30 


0.60 


0.15 


0.01 


0.13 


0.12 


58 


41 


0.12 


0.19 


0.63 


0.31 


0.62 


0.15 


0.01 


0.14 


0.12 


56 


46 


0.10 


0.21 


0.65 


0.32 


0.63 


0.16 


0.00 


0.14 


0.12 


54 


48 


0.09 


0.22 


0.66 


0.32 


0.64 


0.16 


0.00 


0.14 


012 


52 


50 


0.03 0.22 


0.66 


0.32 


0.64 


0.16 0.00 


0.14 0.11 


50 



88 TABLE LXVIII. 

Moon's Horary Motion in Longitude. 
Argument. Argument of the Evection. 





o 


Is 


m 


111^ 


IV* 


Ys | 


o 



80.3 


74.7 


59.6 


39.4 


19.8 


5.9 


o 
30 


1 


80.3 


74.3 


58.9 


38.7 


19.3 


5.6 


29 


2 


80.3 


73.9 


58.3 


38.0 


18.7 


5.3 


28 


3 


80.2 


73.5 


57.7 


37.3 


18.1 


5.0 


27 


4 


80.2 


73.1 


57.1 


36.6 


17.6 


4.7 


26 


5 


80.1 


72.7 


56.4 


36.0 


17.0 


4.4 


25 


6 


80.1 


72.3 


55.8 


35.3 


16.5 


4.1 


24 


7 


80.0 


71.9 


55.1 


34.6 


15.9 


3.8 


23 


8 


79.9 


71.4 


54.5 


33.9 


15.4 


3.6^ 


22 


9 


79.8 


71.0 


53.8 


33.2 


14.9 


3.4 


21 


10 


79.7 


70.5 


53.1 


32.5 


14.4 


3.1 


20 


11 


79.5 


70.1 


52.5 


31.9 


13.9 


2.9 


19 


12 


79.4 


69.6 


51.8 


31.2 


13.4 


2.7 


18 


13 


79.2 


69.1 


51.1 


30.5 


12.9 


2.5 


17 


14 


79.1 


68.6 


50.5 


29.9 


12.4 


2.3 


16 


15 


78.9 


68.1 


49.8 


29.2 


11.9 


2.1 


15 


16 


78.7 


67.6 


49.1 


28.6 


11.4 


2.0 


14 


17 


78.5 


67.0 


48.4 


27.9 


11.0 


1.8 


13 


18 


78.2 


66.5 


47.7 


27.2 


10.5 


1.7 


12 


19 


78.0 


66.0 


47.0 


26.6 


10.1 


1.6 


11 


20 


77.8 


65.4 


46.4 


26.0 


9.7 


1.4 


10 


21 


77.5 


64.9 


45.7 


25.3 


9.3 


1.3 


9 


22 


77.2 


64.3 


45.0 


24.7 


8.8 


1.2 


8 


23 


77.0 


63.7 


44.3 


24.1 


8.4 


1.2 


7 


24 


76.7 


63.2 


43.6 


23.5 


8.0 


1.1 


6 


25 


76.4 


62.6 


42.9 


22.8 


7.7 


1.0 


5 


26 


76.1 


62.0 


42.2 


22.2 


7.3 


1.0 


4 


27 


75.7 


61.4 


41.5 


21.6 


6.9 


0.9 


3 


28 


75.4 


60.8 


40.8 


21.0 


6.6 


0.9 


2 


29 


75.0 


60.2 


40.1 


20.4 


6.2 


0.9 


1 


30 


74.7 


59.6 


39.4 


19.8 


5.9 


0.9 





i xis 


X* 


IX* 


VIII* 


VII* 


VI* 





Arguments. 



TABLE LXIX. 

Moon's Horary Motion in Longitude. 

Sum of Equations, 2, 3, <fec, and Evection corrected. 



1 


0" 


10" 


20" 


1 




I 
II 

III 

IV 

V 

VI 


o 










0.0 
0.0 
0.1 

0.2 
0.3 
0.4 
0.5 


0.2 
0.2 
0.2 

0.2 
0.2 
0.2 
0.2 


0.5 
0.4 
0.3 

0.2 
0.1 
0.0 
0.0 


s o 1 
XII 
XI 
X 

IX 
VIII o 1 
VII 
VI o 1 




0" 


10" 


20" 


1 

1 



TABLE LXX. 



80 



Moon's Horary Motion in Longitude. 
Arguments. Sum of preceding equations, and Anomaly corrected. 



I 


0" 


10" 


20" 


30" 


40" 


50" 


60" 


70" 


80" 


90" 


100" 




s o 



4.1 


5.3 


6.5 


7.6 


8.8 


10.0 


11.2 


12.4 


13.5 


14.7 


15.9 


s o 

XII 


5 


4.1 


5.3 


6.5 


7.7 


8.8 


10.0 


11.2 


12.3 


13.5 


14.7 


15.9 


25 


10 


4.2 


5.4 


6.5 


7.7 


8.8 


10.0 


11.2 


12.3 


13.5 


14.6 


15.8 


20 


15 


4.3 


5.5 


6.6 


7.7 


8.9 


10.0 


11.1 


12.3 


13.4 


14.5 


15.7 


15 


20 


4.5 


5.6 


6.7 


7.8 


8.9 


10.0 


11.1 


12.2 


13.3 


14.4 


15.5 


10 


25 


4.8 


5.8 


6.9 


7.9 


9.0 


10.0 


11,0 


12.1 


13.1 


14.2 


15.2 


5 


I 


5.1 


6.0 


7.0 


8.0 


9.0 


10.0 


11.0 


12.0 


13.0 


14.0 


14.9 


XI 


5 


5.4 


6.3 


7.2 


8.2 


9.1 


10.0 


10.9 


11.8 


12.8 


13.7 


14.6 


25 


10 


5.7 


6.6 


7.4 


8.3 


9.2 


10.0 


10.8 


11.7 


12.6 


13.4 


14.3 


20 


15 


6.1 


6.9 


7.7 


8.5 


9.2 


10.0 


10.8 


11.5 


12.3 


13.1 


13.9 


15 


20 


6.6 


7.2 


7.9 


S.6 


9.3 


10.0 


10.7 


11.4 


12.1 


12.8 


13.4 


10 


25 


7.0 


7.6 


8.2 


8.8 


9.4 


10.0 


10.6 


11.2 


11.8 


12.4 


13.0 


5 


II 


7.5 


8.0 


8.5 


9.0 


9.5 


10.0 


10.5 


11.0 


11.5" 


12.0 


12.5 


X 


5 


7.9 


8.4 


8.8 


9.2 


9.6 


10.0 


10.4 


10.8 


11.2 


11.6 


12.1 


25 1 


10 


8.4 


8.7 


9.1 


9.4 


9.7- 


10.0 


10.3 


10.6 


10.9 


11.3 


11.6 


20 I 


15 


8.9 


9.1 


9.4 


9.6 


9.S 


10.0 


10.2 


10.4 


10.6 


10.9 


11.1 


15 


20 


9.4 


9.5 


9.7 


9.8 


9.9 


10.0 


10.1 


10.2 


10.3 


10.5 


10.6 


10 


25 


9.9 


9.9 


9.9 


10.0 


10.0 


10.0 


10.0 


10.0 


10.1 


10.1 


10.1 


5 


ni 


10.4 


10.3 


10.2 


10.1 


10.1 


10.0 


9.9 


9.9 


9.8 


9.7 


9.6 


IX 0! 
1 


5 


10.8 


10.7 


10.5 


10.3 


10.2 


10.0 


9.8 


9.7 


9.5 


9.3 


9.2 


25 


10 


11.3 


11.0 


10.8 


10.5 


10.3 


10.0 


9.7 


9.5 


9.2 


9.0 


8.7 


20 


15 


11.7 


11.4 


11.0 


10.7 


10.3 


10.0 


9.7 


9.3 


9.0 


8.6 


8.3 


15 


20 


12.1 


11.7 


11.3 


10.9 


10.4 


10.0 


9.6 


9.1 


8.7 


8.3 


7.9 


10 


25 


12.5 


12.0 


11.5 


11.0 


10.5 


10.0 


9.5 


9.0 


8.5 


8.0 


7.5 


5 


IV 


12.9 


12.3 


11.7 


11.2 


10.6 


10.0 


9.4 


8.8 


8.3 


7.7 


7.1 


VIII 


5 


13.3 


12.6 


11.9 


11.3 


10.6 


10.0 


9.4 


8.7 


8.1 


7.4 


6.7 


25 


10 


13.6 


12.9 


12.1 


11.4 


10.7 


10.0 


9.3 


8.6 


7.9 


7.1 


6.4 


20 


15 


13.9 


13.1 


12.3 


11.5 


10.8 


10.0 


9.2 


8.5 


7.7 


6.9 


6.1 


15 


20 


14.1 


13.3 


12.5 


11.6 


10.8 


10.0 


9.2 


8.4 


7.5 


6.7 


5.9 


10 


25 


14.4 


13.5 


12.6 


11.7 


10.9 


10.0 


9.1 


8.3 


7.4 


6.5 


5.6 


5 


V 


14.6 


13.7 


12.7 


11.8 


10.9 


10.0 


9.1 


8.2 


7.3 


6.3 


5.4 


VII 


5 


14.7 


13.8 


12.8 


11.9 


10.9 


10.0 


9.1 


8.1 


7.2 


6.2 


5.3 


25 


10 


14.9 


13.9 


12.9 


12.0 


11.0 


10.0 


9.0 


8.0 


7.1 


6.1 


5.1 


20 


15 


15.0 


14.0 


13.0 


12.0 


11.0 


10.0 


9.0 


8.0 


7.0 


6.0 


5.0 


15 


20 


15.1 


14.1 


13.0 


12.0 


11.0 


10.0 


9.0 


8.0 


7.0 


5.9 


4.9 


10 


25 


15.1 


14.1 


13.1 


12.0 


11.0 


10.0 


9.0 


8.0 


6.9 


5.9 


4.9 


5 


VI 


15.1 


14.1 


13.1 


12.1 


11.0 


10.0 
50" 


9.0 

60" 


8.0 
70" 


69 
80" 


5.9 
90" 


4.9 
100" 


VI 


I 


0" 


10" 


20" 


30" 


40" 





90 



TABLE LXXI. 

Moon's Horary Motion in Longitude. 
Argument. Anomaly corrected. 



7 

8 

9 

10 

11 
12 
13 
14 
15 

16 
17 
18 
19 
20 

21 
22 
23 
24 
25 

26 
27 
28 
29 
30 



0* diff. 



441.5 
441.5 
441.3 
441.1 
440.8 
440.4 

439.9 
439.4 

438.7 
438.0 
437.2 

436.3 
435.3 
434.2 
433.1 

431.8 

430.5 
429.1 
427.6 
426.1 
424.5 

422.7 
421.0 
419.1 
417.2 
415.2 

413.1 
410.9 
408.7 
406.4 
404.1 

XI* 



0.0 
0.1 
0.2 
0.3 
0.4 
0.5 

0.5 

0.7 
0.7 
0.8 
0.9 

1.0 
1.1 
1.1 
1.3 
1.3 

1.4 
1.5 
1.5 
1.6 

1.7 

1.7 
1.9 
1.9 
2.0 

2.1 

2.2 
2.2 
2.3 
2.3 



diff. | II* diff Ills diff. IV* diff. V* diff. 



404.1 
401.6 
399.2 
396.6 
394.0 
391.3 

388.6 
385.8 
383.0 
380.1 
377.1 

374.1 
371.1 
368.0 
364.8 
361.6 

358.4 
355.1 
351.8 
348.4 
345.0 

341.6 

338.1 
334.6 
331.1 
327.5 

324.0 
320.3 
316.7 
313.0 
309.3 

X* 



2.5 
2.4 
2.6 
2.6 
2.7 

2.7 

2.8 
2.8 
2.9 
3.0 

3.0 

3.0 
3.1 
3.2 
3.2 
3.2 

3.3 
3.3 
3.4 
3.4 

3.4 

3.5 
3.5 
3.5 

3.6 

3.5 

3.7 
3.6 

3.7 
3.7 



309.3 
305.6 
301.9 
298.1 
294.4 
290.6 

286.8 
283.0 
279.2 
275.4 
271.5 

267.7 
263.8 
260.0 
256.2 
252.3 

248.5 
244.6 
240.8 
236.9 
233.1 

229.3 
225.4 
221.6 
217.8 
214.0 

210.3 
206.5 
202.8 
199.0 
195.3 

IX* 



3.7 

3.7 

3.8 

3.7 

3.8 ! 

3.8 

3.8 
3.8 
3.8 
3.9 
3.8 

3.9 

3.8 
3.8 
3.9 



3.9 

3.8 
3.9 
3.8 
3.8 

3.9 
3.8 
3.8 
3.8 

3.7 

3.8 
3.7 
3.8 
3.7 



195.3 
191.6 
i 187.9 
! 184.3 
180.6 
177.0 



173.4 
169.8 
166.3 
162.8 
159.3 

155.8 
152.4 
148.9 
145.5 
142.2 

138.9 
135.6 
132.3 
129.1 
125.9 

122.7 
119.6 
116.5 
113.4 
110.4 

107.4 

104.5 

101.6 

98.7 

95.8 

VIII* 



3.7 
3.7 
3.6 
3.7 
3.6 
3.6 

3.6 

3.5 
3.5 
3.5 
3.5 

3.4 
3.5 
3.4 
3.3 

3.3 

3.3 
3.3 
3.2 

3.2 

3.2 

3.1 
3.1 
3.1 
3.0 

3.0 

2.9 

2.9 
2.9 
2.9 



95.8 
93.0 
90.2 
87.6 
84.9 
82.3 

79.7 
77.1 
74.6 
72.1 
69.7 

67.3 
65.0 
62.7 
60.4 

58.2 

56.1 
53.9 
51.9 
49.8 
47.9 

45.9 

44.0 

42.2 
40.4 

38.7 

37.0 
35.3 
33.7 
32.1 
30.6 

VII* 



2.8 
2.8 
2.6 
2.7 
2.6 
2.6 

2.6 
2.5 

2.5 
2.4 
2.4 

2.3 

2.3 
2,3 

2.2 
2.1 

2.2 
2.0 
2.1 
1.9 
2.0 

1.9 

1.8 
1.8 
1.7 
1.7 

1.7 
1.6 
1.6 
1.5 



30.6 
29.2 
27.8 
26.4 
25.1 
23.8 

22.6 
21.4 
20.3 
19.2 
18.2 

17.2 
16.3 
15.4 
14.6 
13.8 

13.1 
12.4 
11.8 
11.2 
10.7 

10.2 
9.8 
9.4 
9.1 

8.8 

8.6 
8.4 
8.3 
8.2 

8.2 

VI* 



1.4 
1.4 
1.4 
1.3 
1.3 

1.2 

1.2 
1.1 
1.1 
1.0 

1.0 

0.9 

0.9 
0.8 
0.8 

0.7 

0. 
0.6 
0.6 
0.5 

0.5 

0.4 

0.3 
0.3 

0.2 

0.2 
0.1 
0.1 
0.0 



30 
29 
28 
27 
26 
25 

24 
23 
22 
21 
20 

19 
18 
17 
16 
15 

14 
13 
12 
11 
10 

9 

8 
7 



TABLE LXXII. 

Moons Horary Motion in Longitude. 
Arguments. Sum of preceding Equations, and Arg. of Variation. 



91 







tl 




/t 


„ 


,, 


„ 


„ 


„ 






„ 


„ 









50 


100 


150 


200 


250 


300 


350 


400 


450 


500 


550 


600 




s ° 




4.5 


5.5 


6.5 


7.6 


8.6 


9.6 


10.6 


11.6 


12.6 


13.7 


14.7 


15.7 


16.7 


s o 

XII 


5 


4.6 


5.6 


6.6 


7.6 


8.6 


9.6 


10.6 


11.6 


12.6 


13.6 


14.6 


15.6 


16.6 


25 


10 


4.8 


5.8 


6.8 


7.7 


8.7 


9.6 


10.6 


11.5 


12.5 


13.4 


14.4 


15.3 


16.3 


20 


15 


5.3 


6.1 


7.0 


7.9 


8.8 


9.7 


10.5 


11.4 


12.3 


13.1 


14.0 


14.9 


15.8 


15 


20 5.8 


6.6 


7.4 


8.2 


8.9 


9.7 


10.5 


11.2 


12.0 


12.8 


13.5 


14.3 


15.1 


10 


25 6.6 


7.2 


7.8 


8.5 


9.1 


9.7 


10.4 


11.0 


11.7 


12.3 


12.9 


13.6 


14.2 


5 


I 


7.4 


7.8 


8.3 


8.8 


9.3 


9.8 


10.3 


10.8 


11.3 


11.8 


12.3 


12.7 


13.2 


XI 


5 


8.3 


8.6 


8.9 


9.2 


9.5 


9.9 


10.2 


10.5 


10.8 


11.2 


11.5 


11.8 


12.1 


25 


10 


9.2 


9.3 


9.5 


9.6 


9.8 


9.9 


10.1 


10.2 


10.4 


10.5 


10.7 


10.8 


11.0 


20 


15 


10.2 


10.1 


10.1 


10.1 


10.0 


10.0 


10.0 


10.0 


9.9 


9.9 


9.9 


9.8 


9.8 


15 


20 


11.1 


10.9 


10.7 


10.5 


10.3 


10.1 


9.9 


9.7 


9.5 


9-2 


9.0 


8.8 


8.6 


10 


25 


12.1 


11.7 


11.3 


10.9 


10.5 


10.2 


9.8 


9.4 


9.0 


8.6 


8.3 


7.9 


7.5 


5 


II 


12.9 


12.4 


11.8 


11.3 


10.8 


10.2 


9.7 


9.1 


8.6 


8.1 


7.5 


7.0 


6.4 


X 


5 


13.7 


13.0 


12.3 


11.6 


11.0 


10.3 


9.6 


8.9 


8.2 


7.5 


6.9 


6.2 


5.5 


25 


10 


14.3 


13.5 


12.7 


11.9 


11.1 


10.3 


9.5 


8.7 


7.9 


7.1 


6.3 


5.5 


4.7 


20 


15 


14.9 


14.0 


13.1 


12.2 


11.3 


10.4 


9.5 


8.6 


7.7 


6.8 


5.8 


4.9 


4.0 


15 


20 


15.3 


14.3 


13.3 


12.3 


11.4 


10.4 


9.4 


8.4 


7.5 


6.5 


5.5 


4.5 


3.6 


10 


25 


15.5 


14.5 


13.5 


12.4 


11.4 


10.4 


9.4 


8.4 


7.4 


6.3 


5.3 


4.3 


3.3 


5 


III 


15.6 


14.5 


13.5 


12.5 


11.4 


10.4 


9.4 


8.4 


7.3 


6.3 


5.3 


4.2 


3.2 


IX 


5 


15.4 


14.4 


1*4 


12.4 


11.4 


10.4 


9.4 


8.4 


7.4 


6.4 


5.4 


4.4 


3.3 


25 


10 


15.2 


14.2 


13.3 


12.3 


11.3 


10.4 


9.4 


8.5 


7.5 


6.5 


5.6 


4.6 


3.6 


20 


15 


14.8 


13.9 


13.0 


12.1 


11.2 


10.4 


9.5 


8.6 


7.7 


6.8 


5.9 


5.1 


4.2 


15 


20 


14.2 


13.4 


12.6 


11.9 


11.1 


10.3 


9.5 


8.8 


8.0 


7.2 


6.4 


5.6 


4.9 


10 


25 


13.5 


12.9 


12.2 


11.6 


10.9 


10.3 


9.6 


9.0 


8.4 


7.6 


7.0 


6.3 


5.7 


5 


IV 


12.7 


12.2 


11.7 


11.2 


10.7 


10.2 


9.7 


9.2 


8.7 


8.2 


7.7 


7.2 


6.7 


vino 


5 


11.9 


11.5 


11.2 


10.8 


10.5 


10.1 


9.8 


9.5 


9.1 


8.8 


8.4 


8.1 


7.7 


25 


10 


10.9 


10.7 


10.6 


10.4 


10.2 


10.1 


9.9 


9.7 


9.6 


9.4 


9.2 


9.1 


8.9 


20 


15 


9.9 


9.9 


10.0 


10.0 


10.0 


10.0 


10.0 


10.0 


10.0 


10.0 


10.1 


10.1 


10.1 


15 


20 


8.9 


9.1 


9.3 


9.5 


9.7 


9.9 


10.1 


10.3 


10.5 


10.7 


10.9 


11.1 


11.8 


10 


25 


8.0 


8.4 


8.7 


9.1 


9.5 


9.9 


10.2 


10.6 


11.0 


11.3 


11.7 


12.1 


12.5 


5 


V 


7.1 


7.6 


8.2 


8.7 


9.2 


9.8 


10.3 


10.9 


11.4 


11.9 


12.5 


13.0 


13.6 


VII 0, 


5 


6.3 


7.0 


7.6 


8.3 


9.0 


9.7 


10.4 


11.1 


11.8 


12.5 


13.2 


13.9 


14.6 


25 


10 


5.6 


6.4 


7.2 


8.0 


8.8 


9.7 


10.5 


11.3 


12.1 


13.0 


13.8 


14.6 


15.4 


20| 


15 


5.0 


5.9 


6.8 


7.8 


8.7 


9.6 


10.6 


11.5 


12.4 


13.3 


14.3 


15.2 


16.1 


15 


20 


4.6 


5.6 


6.6 


7.6 


8.6 


9.6 


10.6 


11.6 


12.6 


13.6 


14.6 


15.7 


16.7 


10 


25 


4.3 


5.4 


6.4 


7.5 


8.5 


9.6 


10.6 jll. 7 


12.7 


13.8 


14.9 


15.9 


17.0 


5 


VI 


4.2 


5.3 


6.4 


7.4 


8.5 


9.6 


10.6;11.7 


12.8 


13.9 


14.9 


16.0 


17.1 


VI 




,. 


t. 


// 


/. 


„ 


,, 


~T~i~ 


~7,~ 


„ 


// 


// 


" 









50 


100 


150 


200 


250 


300 : 350 


400 


450 


500 


550 


600 





TABLE LXXIII. 

Moon's Horary Motion in Longitude. 
Argument. Argument of the Variation. 



i 0s 


Is 


lis 


Ills 


IV* 


Vs 




o 


„ 


„ 


rr 


// 


// 


„ 


o 





77.2 


57.8 


20.3 


2.4 


21.5 


59.7 


30 


1 


77.2 


56.7 


19.2 


2.5 


22.7 


60.9 


29 


2 


77.1 


55.5 


18.1 


2.6 


23.8 


62.0 


28 


3 


77.0 


54.3 


17.0 


2.7 


25.0 


63.1 


27 


4 


76.8 


53.1 


16.0 


2.9 


26.2 


64.2 


26 


5 


76.6 


51.8 


15.0 


3.1 


27.5 


65.3 


25 


6 


76.4 


50.5 


14.1 


3.3 


28.7 


66.3 


24 


7 


76.1 


49.3 


13.2 


3.7 


30.0 


67.3 


23 


8 


75.7 


48.0 


12.3 


4.0 


31.3 


68.3 


22 


9 


75.3 


46.7 


11.4 


4.4 


32.6 


69.2 


21 


10 


74.9 


45.4 


10.6 


4.9 


33.9 


70.1 


20 


11 


74.4 


44.1 


9.8 


5.3 


35.2 


70.9 


19 


12 


73.9 


42.8 


9.0 


5.9 


36.5 


71.7 


18 


13 


73.3 


41.5 


8.3 


6.4 


37.8 


72.5 


17 


14 


72.7 


40.2 


7.6 


7.0 


39.2 


73.3 


16 


15 


72.0 


38.9 


7.0 


7.7 


40.5 


74.0 


15 


16 


71.3 


37.5 


6.4 


8.3 


41.8 


74.7 


14 


17 


70.6 


36.2 


5.8 


9.1 


43.2 


75.3 


13 


18 


69.8 


34.9 


5.3 


9.8 


44.5 


75.8 


12 


19 


69.0 


33.6 


4.8 


10.6 


45.8 


76.4 


11 


20 


68.1 


32.3 


4.4 


11.5 


47.2 


76.9 


10 


21 


67.2 


31.1 


4.0 


12.3 


48.5 


1 77.3 


9 


22 


66.3 


29.8 


3.7 


13.2, 


49.8 


77.7 


8 


23 


65.3 


28.6 


3.3 


14.2 


51.1 


78.1 


7 


24 


64.4 


27.3 


3.1 


15.1 


52.4 


78.4 


6 


25 


63.4 


26.1 


2.9 


16.1 


53.6 


78.6 


5 


26 


62.3 


24.9 


2.7 


17.1 


54.9 


78.9 


4 


27 


61.2 


23.7 


2.5 


18.2 


56.1 


79.0 


3 


28 


60.1 


22.5 


2.5 


19.3 


57.3 


79.2 


2 


| 29 


59.0 


21.4 


2.4 


20.4 


58.5 


79.2 


1 


] 30 


57.8 


20.3 


2.4 


21.5 


59.7 


79.2 







XI" 


X* 


IX* 


vni* 


YTLs 


VI? 


* 



TABLE LXXIV. 93 

Moon's Horary Motion in Longitude. 
Arguments. Arg. of Reduction and Sum of preceding Equations 





„ 


„ 


,, 


„ 


„ 


„ | „ 


// i ,/ 


„ 


,r | „ 


„ 


„ 









50 


100 


150 


200 


250 


300 


350 400 


450 


500 550 

I 


600 


650 




8 ° 


// 


,/ 


ft 


// 


// 


// 


>, 


'/ 


" 


// 


// 


// 


// 


" 


s ° 





3.3 


3.1 


2.9 


2.7 


2.5 


2.3 


2.1 


1.9 


1.7 


1.5 


1.3 


1.1 


0.9 


0.7 


XII 


r> 


3.3 


3.1 


2.9 


2.7 


2.5 


2.3 


2.1 


1.9 


1.7 


1.5 


1.3 


1.1 


0.9 


0.7 


25 


10 


3.2 


3.0 


2.8 


2.6 


2.4 


2.3 


2.1 


1.9 


1.7 


1.5 


1.3 


1.1 


1.0 


0.8 


20 


15 


3.1 


2.9 


2.8 


2.6 


2.4 


2.2 


2.1 


1.9 


1.7 


1.5 


1.4 


1.2 


1.0 


0.9 


15 


20 


3.0 


2.8 


2.7 


2.5 


2.4 


2.2 


2.1 


1.9 


1.8 


1.6 


1.5 


1.3 


1.1 


1.0 


10 


25 


2.8 


2.7 


2.6 


2,4 


2.3 


2.2 


2.1 


1.9 


1.8 


1.7 


1.5 


1.4 


1.3 


1.2 


5 


I 


2.6 


2.5 


2.4 


2.3 


2.2 


2.1 


2.0 


1.9 


1.8 


1.7 


1.6 


1.5 


1.4 


1.3 


XI 


5 


2.4 


2.4 


2.3 


2.2 


2.2 


2.1 


2.0 


2.0 


1.9 


1.8 


1.8 


1.7 


1.6 


1.6 


25 


10 


2.2 


2.2 


2.2 


2.1 


2.1 


2.0 


2.0 


2.0 


1.9 


1.9 


1.9 


1.8 


1.8 


1.8 


20 


15 


2.0 


2.0 


2.0 


2.0 


2.0 


2.0 


2.0 


2.0 


2.0 


2.0 


2.0 


2.0 


2.0 


2.0 


15 


20 


1.8 


1.8 


1.8 


1.9 


1.9 


1.9 


2.0 


2.0 


2.1 


2.1 


2.1 


2.2 


2.2 


2.2 


10 


25 


1.6 


1.6 


1.7 


1.8 


1.8 


1.9 


2.0 


2.0 


2.1 


2.2 


2.2 


2.3 


2.4 


2.4 


5 


II 


1.4 


1.5 


i.6 


1.7 


1.8 


1.9 


2.0 


2.1 


2.2 


2.3 


2.4 


2.5 


2.6 


2.7 


X 


5 


1.2 


1.3 


1.4 


1.6 


1.7 


1.8 


1.9 


2.1 


2.2 


2.3 


2.5 


2.6 


2.7 


2.8 


25 


10 


1.0 


1.2 


1.3 


1.5 


1.6 


1.8 


1.9 


2.1 


2.2 


2.4 


2.5 


2.7 


2.9 


3.0 


20 


15 


0.9 


1.1 


1.2 


1.4 


1.6 


1.8 


1.9 


2.1 


2.3 


2.5 


2.6 


2.8 


3.0 


3.1 


15 


20 


0.8 


1.0 


1.2 


1.4 


1.6 


1.7 


1.9 


2.1 


2.3 


2.5 


2.7 


2.9 


3.0 


3.2 


10 


25 


0.7 


0.9 


1.1 


1.3 


1.5 


1.7 


1.9 


2.1 


2.3 


2.5 


2.7 


2.9 


3.1 


3.3 


5 


III 


0.7 


0.9 


1.1 


1.3 


1.5 


1.7 


1.9 


2.1 


2.3 


2.5 


2.7 


2.9 


3.1 


3.3 


IX 


5 


0.7 


0.9 


1.1 


1.3 


1.5 


1.7 


1.9 


2.1 


2.3 


2.5 


2.7 


2.9 


3.1 


3.3 


25 


10 


0.8 


1.0 


i.2 


1.4- 


1.6 


1.7 


1.9 


2.1 


2.3 


2.5 


2.7 


2.9 


3.0 


3.2 


20 


15 


0.9 


1.1 


1.2 


1.4 


1.6 


1.8 


1.9 


2.1 


2.3 


2.5 


2.6 


2.8 


3.0 


3.1 


15 


20 


1.0 


1.2 


1.3 


1.5 


1.6 


1.8 


1.9 


2.1 


2.2 


2.4 


2.5 


2.7 


2.9 


3.0 


10 


25 


1.2 


1.3 


1.4 


1.6 


1.7 


1.8 


1.9 


2.1 


2.2 


2.3 


2.5 


2.6 


2.7 2.8 


5 


IV 


1.4 


1.5 


1.6 


1.7 


1.8 


1.9 


2.0 


2.1 


2.2 


2.3 


2.4 


2.5 


2.6 2.7 


VIII 


5 


1.6 


1.6 


1.7 


1.8 


1.8 


1.9 


2.0 


2.0 


2.1 


2.2 


2.2 


2.3 


2.4 2.4 


25 


10 


1.8 


1.8 


1.8 


1.9 


1.9 


1.9 


2.0 


2.0 


2.1 


2.1 


2.1 


2.2 


2.2|2.2 


20 


15 


2.0 


2.0 


2.0 


2.0 


2.0 


2.0 


2.0 


2.0 


2.0 


2.0 


2.0 


2.0 


2.0 


2.0 


15 


20 


2.2 


2.2 


2.2 


2.1 


2.1 


2.0 


2.0 


2.0 


1.9 


1.9 


1.9 


1.8 


1.8 


1.8 


10 


25 


2.4 


2.4 


2.3 


2.2 


2.2 


2.1 


2.G 


2.0 


1.9 


1.8 


1.8 


1.7 


1.6 


1.6 


5 


V 


2.6 


2.5 


2.4 


2.3 


2.2 


2.1 


2.0 


1.9 


1.8 


1.7 


1.6 


1.5 


1.4 


1.3 


Vll o 


5 


2.8 


2.7 


2.6 


2.4 


2.3 


2.2 


2.1 


1.9 


1.8 


1.7 


1.5 


1.4 


1.3 


1.2 


25 


10 


3.0 


2.8 


2.7 


2.5 


2.4 


2.2 


2.1 


1.9 


1.8 


1.6 


1.5 


1.3 


1.1 


1.0 


20 


15 


3.1 


2.9 


2.8 


2.6 


2.4 


2.2 


2.1 


1.9 


1.7 


1.5 


1.4 


1.2 


1.0 


0.9 


15 


20 


3.2 


3.0 


2.8 


2.6 


2.4 


2.3 


2.1 


1.9 


1.7 


1.5 


1.3 


1.1 


1.0 


0.8 


10 


25 


3.3 


3.1 


2.9 


2.7 


2.5 


2.3 


2.1 


1.9 


1.7 


1.5 


1.3 


1.1 


0.9 


0.7 


5 


VI 


3.3 


3.1 


2.9 


2.7 


2.5 


2.3 


2.1 


1.9 


1.7 


1.5 


1.3 


1.1 


0.9 


0.7 


VI 







50 


100 


150 


200 


250 


300 


350 


400 


450 


500 


550 


600 


650 





94 TABLE LXXV. 

Moon's Horary Motion in Long. 
Arg. Arg. of Reduction. 



TABLE LXXVI. 

Moon's Horary Motion in Long, 

(Equation of the second order.) 

Arguments. Arg's.of Table LXX, 





Os Vis 


Is VI's 


Us VIIIs 




o 


n 


» 


» 


o 





2.1 


6.0 


14.0 


30 


1 


2.1 


6.3 


14.2 


29 


2 


2.1 


6.5 


14.4 


28 


3 


2.1 


6.8 


14.7 


27 


4 


2.2 


7.0 


14.9 


26 


5 


2.2 


7.3 


15.1 


25 


6 


2.2 


7.5 


15.3 


24 


7 


2.3 


7.8 


15.5 


23 


8 


2.4 


8.1 


15.7 


22 


9 


2.5 


8.4 


15.9 


21 


10 


2.5 


• 8.6 


16.1 


20 


11 


2.6 


8.9 


16.2 


19 


12 


2.7 


9.2 


16.4 


18 


13 


2.9 


9.4 


16.6 


17 


14 


3.0 


9.7 


16.7 


16 


15 


3.1 


10.0 


16.9 


15 


16 


3.3 


10.3 


17.0 


14 


17 


3.4 


10.6 


17.1 


13 


18 


3.6 


10.8 


17.3 


12 


19 


3.8 


11.1 


17.4 


11 


20 


3.9 


11.4 


17.5 


10 


21 


4.1 


11.6 


17.5 


9 


22 


4.3 


11.9 


17.6 


8 


23 


4.5 


12.2 


17.7 


7 


24 


4.7 


12.5 


17.8 


6 


25 


4.9 


12.7 


17.8 


5 


26 


5.1 


13.0 


17.8 


4 


27 


5.3 


13.2 


17.9 


3 


28 


5.6 


13.5 


17.9 


2 


29 


5.8 


13.7 


17.9 


1 


30 


6.0 


14.0 


17.9 







XTs Vs 


XsIVs 


lXsIHs 





Constant to be added 27'24".0. 

TABLE LXXVII. 
Moon's Horary Motion in Longitude. 
(Equations of the second order.) 
Arguments. Arguments of Tables LXXII and LXXIV. 







Variation. 


Reduction. 


tr 


rr 


,, 


„ 


,, 


„ 


rr 


„ 


rr 









100 


200 


300 


400 


500 


600 





600 


8 


s ° 


rr 


" 


// 


„ 


// 


,r 


rr 


// 


0. 


VI. 


0.14 


0.14 


0.14 


0.14 


0.14 


0.1 A 


0.14 


03 


0.03 


I. 


VII. 


0.22 


0.19 


0.16 


0.13 


0.10 


0.06 


0.02 


0.01 


0.05 


I. 


VII. 15 


0.23 


0.20 


0.17 


0.13 


0.10 


0.05 


0.01 


0.01 


0.06 


II. 


VIII. 


0.22 


0.19 


0.16 


0>3 


0.10 


0.07 


0.03 


0.01 


0.05 


III. 


IX. 


0.14 


0.14 


0.14 


0.14 


0.14 


0.14 


0.14 


0.03 


0.03 


IV. 


X. 


0.06 


0.09 


0.12 


0.15 


0.18 


0.21 


0.26 


0.05 


0.01 


IV. 


X. 15 


0.05 


0.08 


0.11 


0.15 


0.18 


0.23 


0.28 


0.05 


0.00 


V. 


XI. 


0.06 


0.09 


0.12 


0.15 


0.18 


0.22 


0.26 


0.05 


0.01 


VI. 


XII. 


0.14 


0.14 


0.14 


0.14 


0.14 


C.14 


0.14 


0.03 


0.03 







rr 


'/ 


\ 


Arg 







50 


100 


s 



o 



0.05 


0.05 


0.05 


I 





0.08 


0.05 


0.02 


n 





0.10 


0.05 


0.00 


III 





0.10 


0.05 


0.00 


IV 





0.09 


0.05 


0.01 


V 





0.07 


0.05 


0.03 


VI 





0.05 


0.05 


0.05 


VII 





0.03 


0.05 


0.07 


VIII 





0.01 


0.05 


0.09 


IX 





0.00 


0.05 


0.10 


X 





0.00 


0.05 


0.10 


XI 





0.02 


0.05 


0.08 


XII 





0.05 


0.05 


0.05 






rr 


rr 


rr 









50 


100 



TABLE LXXVIII. 

Mooris Horary Motion in Longitude. 

(Equations of the second order.) 

Arguments. Args. of Evection, Anomaly, Variation, Reduction, 



95 





Evec. 


Anom. 


Var. 


Red. 


1 Evec. 


Anom. 


Var. 


Red. | 





0.16 


1.05 


0.34 


0.08 


0.16 


rr 

1.05 


0.34 


0.08 


XII 


5 


0.15 


0.93 


0.28 


0.09 


0.18 


1.17 


0.40 


0.06 


25 


10 


0.13 


0.81 


0.22 


0.10 


0.19 


1.28 


0.46 


0.05 


20 


15 


0.12 


0.70 


0.17 


0.11 


0.21 


1.40 


0.51 


0.04 


15 1 


20 


0.10 


0.59 


0.12 


0.12 


0.22 


1.50 


0.56 


0.03 


10 


25 


0.09 


0.49 


0.08 


0.13 


0.24 


1.60 


0.60 


0.02 


5 


I 


0.08 


0.40 


0.05 


0.14 


0.25 


1.70 


0.63 


0.01 


XI 


5 


0.07 


0.31 


0.02 


0.15 


0.26 


1.78 


0.66 


0.01 


25 


10 


0.05 


0.24 


0.01 


0.15 


0.27 


1.86 


0.67 


0.00 


20 


15 


0.04 


0.17 


0.01 


0.15 


0.28 


1.92 


0.67 


0.00 


15 


20 


0.03 


0.12 


0.01 


0.15 


0.29 


1.98 


0.67 


0.00 


10 


25 


0.03 


0.07 


0.03 


0.15 


0.30 


2.02 


0.65 


0.01 


5 


II 


0.02 


0.04 


0.06 


0.14 


0.31 


2.05 


0.62 


0.01 


X 


5 


0.01 


0.02 


0.09 


0.13 


0.32 


2.08 


0.59 


0.02 


25 


10 


0.01 


0.00 


0.13 


0.12 


0.32 


2.09 


0.54 


0.03 


20 


15 


0.00 


0.00 


0.18 


0.11 


0.32 


2.10 


0.50 


0.04 


15 


20 


0.00 


0.00 


0.24 


0.10 


0.33 


2.09 


0.44 


0.05 


10 


25 


0.00 


0.02 


0.29 


0.09 


0.33 


2.08 


0.39 


0.06 


5 


III 


0.00 


0.04 


0.35 


0.08 


0.33 


2.06 


0.33 


0.08 


IX 


5 


0.00 


0.07 


0.40 


0.06 


0.33 


2.03 


0.27 


0.09 


25 


10 


0.01 


0.10 


0.46 


0.05 


0.32 


2.00 


0.22 


0.10 


20 


15 


0.01 


0.14 


0.51 


0.04 


0.32 


1.96 


0.17 


0.11 


15 


20 


001 


0.18 


0.56 


0.03 


0.31 


1.91 


0.12 


0.12 


10 


25 


0.02 


0.23 


0.60 


0.02 


0.31 


1.87 


0.08 


0.13 


5 


IV 


0.03 


0.28 


0.63 


0.01 


0.30 


1.82 


0.05 


0.14 


VIII o 


5 


0.03 


0.34 


0.66 


0.01 


0.29 


1.76 


0.02 


0.15 


25 


10 


0.04 


0.39 


0.67 


0.00 


0.28 


1.70 


0.01 


0.15 


20 


15 


0.05 


0.45 


0.68 


0.00 


0.27 


1.64 


0.00 


0.15 


15 


20 


0.06 


0.52 


0.67 


0.00 


0.26 


1.58 


0.00 


0.15 


10 


25 


0.08 


0.58 


0.66 


0.01 


0.25 


1.52 


0.02 


0.15 


5 


V 


0.09 


0.64 


0.64 


0.01 


0.24 


1.45 


0.04 


0.14 


VII o 


5 


0.10 


0.71 


0.60 


0.02 


0.23 


1.39 


0.08 


0.13 


25 


10 


0.11 


0.78 


0.56 


0.03 


0.22 


1.32 


0.12 


0.12 


20 


15 


0.12 


0.84 


0.51 


0.04 


0.20 


1.25 


0.16 


0.11 


16 


20 


0.14 


0.91 


0.46 


0.05 


0.19 


1.18 


0.22 


0.10 


10 


25 


0.15 


0.98 


0.40 


0.06 


0.18 


1.12 


0.28 


0.09 


5 


VI 


0.16 


1.05 


0.34 


0.08 


0.16 


1.05 


0.34 


0.08 


VI j 



90 



TABLE LXXIX. 

Moon's Horary Motion in Latitude. 
Argument. Arg. I of Latitude. 





0* 


Is 


II* 


III* 


IV* 


V* 




o 



378.0 


354.3 


289.2 


200.0 


110.8 


45.7 


o 
30 


1 


378.0 


352.7 


286.5 


196.9 


108.1 


44.2 


29 


2 


377.9 


351.1 


283.8 


193.8 


105.4 


42.7 


28 


3 


377.8 


349.4 


281.0 


190.7 


102.8 


41.3 


27 


4 


377.6 


347.7 


278.3 


187.5 


100.2 


39.9 


26 


5 


377.3 


346.0 


275.5 


184.4 


97.7 


38.6 


25 


6 


377.0 


344.2 


272.6 


181.3 


95.1 


37.3 


24 


7 


376.7 


342.3 


269.8 


178.2 


92.6 


36.1 


23 


8 


376.3 


340.5 


266.9 


175.1 


90.2 


34.9 


22 


9 


375.8 


338.5 


264.0 


172.1 


87.7 


33.8 


21 


10 


375.3 


336.6 


261.1 


169.0 


85.3 


32.7 


20 


11 


374.7 


334.5 


258.1 


165.9 


83.0 


31.6 


19 


12 


374.1 


332.5 


255.2 


162.9 


80.7 


30.7 


18 


13 


373.5 


330.4 


252.2 


159.8 


78.1 


29.7 


17 


14 


372.7 


328.3 


249.2 


156.8 


76.1 


28.9 


16 


15 


372.0 


326.1 


246.2 


153.8 


73.9 


28.0 


15 


16 


371.1 


323.9 


243.2 


150.8 


71.7 


27.3 


14 


17 


370.3 


321.9 


240.2 


147.8 


69.6 


26.5 


13 


18 


369.3 


319.3 


237.1 


144.8 


67.5 


25.9 


12 


19 


368.4 


317.0 


234.1 


141.9 


65.5 


25.3 


11 


20 


367.3 


314.7 


231.0 


138.9 


63.4 


24.7 


10 


21 


366.2 


312.3 


227.9 


136.0 


61.5 


24.2 


9 


22 


365.1 


309.S 


224.9 


133.1 


59.5 


23.7 


8 


23 


363.9 


307.4 


221.8 


130.2 


57.7 


23.3 


7 


24 


362.7 


304.9 


218.7 


127.4 


55.8 


23.0 


6 


25 


361.4 


302.3 


215.6 


124.5 


54.0 


22.7 


5 


26 


360.1 


299.8 


212.5 


121.7 


52.3 


22.4 


4 


27 


358.7 


297.2 


209.3 


119.0 


50.6 


22.2 


3 


28 


357.3 


294.6 


206.2 


116.2 


48.9 


22.1 


2 


29 


355.8 


291.9 


203.1 


113.5 


47.3 


22.0 


1 


30 


354.3 


289.2 


200.0 


110.8 


45.7 


22.0 







XI* 


X* 


IX* 


VIII* 


VII* 


Yh 





TABLE LXXX. 

Moon's Horary Motion in Latitude. 
Arguments. Args. V, VI, VII, VIII, IX, X, XI, and XII, of Latitude 



Arg. 


V 1 VI 


VII 


VIII 


IX 


X 


XI 


XII 


Arg. 





0.00 0.50 


0.34 


0.00 


0.50 


0.04 


0.12 


0.08 


1000 


50 


0.01 0.49 


0.33 


0.00 


0.49 


0.04 


0.12 


0.07 


950 


100 


0.04 0.45 


0.30 


0.02 


0.45 


0.04 


0.11 


0.05 


900 


150 


0.09 0.40 


0.27 


0.04 


0.40 


0.03 


0.10 0.03 


850 


200 


0.16 0.33 


0.22 


0.06 


0.33 


0.03 


0.08 0.01 


800 


250 


0.23 0.25 


0.17 


0.09 


0.25 


0.02 


0.06 0.00 


750 


300 


0.30 0.17 


0.12 


0.12 


0.17 


0.01 


0.040.01 


700 


350 


0.37 0.10 


0.07 


0.14 


0.10 


0.01 


0.020.03 


650 


400 


0.42 0.05 


0.04 


0.16 


0.05 


0.00 


0.010.05 


600 


450 


0.45 0.01 


0.01 


0.18 


0.01 


0.00 


0.00,0.07 


550 


500 


0.46 0.00: 0.00 0.18 


0.00 


0.00 


0.00|0.08 


500 



TABLE LXXXI. Moon J s Horary Motion in Latitude. 97 

Arguments. Preceding equation, and Sum of equations of Horary 

Motion in Longitude, except the last two. 



Pr. 

eq. 

20 


0" 


50" 


100" 


150" 


200" 


250" 


300" 


350" 


400" 


£50" 


500" 


550" 


600" 


650" 




1".6 
59.0 


1".4 
54.5 


l."l 
50.0 


0".9 

45.4 


0".6 
40.9 


0".4 


0".l 


0\2 
27.3 


0".4 

22.8 


0".7 
18.2 


0".9 
13.7 


1".2 
9.1 


1".4 
4.6 


1".7 

0.1 


Diff. 
4.5 


36.4 


31.8 


30 


57.4 


53.1 


4S.9 


44.6 


40.3 


36.0 


31.7 


27.4 


23.2 


18.9 


14.6 


10.3 


6.0 


1.7 


4.3 


40 


55.8 


51.8 


47.7 


43.7 


39.7 


35.6 


31.6 


27.6 


23.6 


19.5 


15.5 


11.5 


7.4 


3.4 


4.0 


50 


54.2 


50.4 


46.6 


42.9 


39.1 


35.3 


31.5 


27.7 


24.0 


20.2 


16.4 


12.6 


8.8 


5.1 


3.8 


60 


52.6 


49.1 


45.5 


42.0 


38.5 


34.9 


31.4 


27.9 


24.4 


20.8 


17.3 


13.8 


10.2 


6.7 


3.5 


70 


51.0 


47.7 


44.4 


41.1 


37.9 


34.6 


31.3 


28.0 


24.8 


21.5 


18.2 


14.9 


11.7 


8.4 


3.3 


80 


49.3 


46.3 


43.3 


40.3 


37.3 


34.2 


31.2 


28.2 


25.2 


22.1 


19.1 


16.1 


13.1 


10.0 


3.0 


90 


47.7 


45.0 


42.2 


39.4 


36.7 


33.9 


31.1 


28.3 


25.6 


22.8 


20.0 


17.3 


14.5 


11.7 


2.8 


100 


46.1 


43.6 


41.1 


38.6 


36.0 


33.5 


31.0 


28.5 


26.0 


23.4 


20.9 


18.4 


15.9 


13.4 


2.5 


110 


44.5 


42.2 


40.0 


37.7 


35.4 


33.2 


30.9 


28.6 


26.4 


24.1 


21.8 


19.6 


17.3 


15.0 


2.3 


120 


42.9 


40.9 


38.9 


36.9 


34.8 


32.8 


30.8 


28.8 


26.8 


24.8 


22.7 


20.7 


18.7 


16.7 


2.0 


130 


41.3 


39.5 


37.8 


36.0 


34.2 


32.5 


30.7 


28.9 


27.2 


25.4 


23.7 


21.9 


20.1 


18.4 


1.8 


140 


39.7 


38.2 


36.7 


35.1 


33.6 


32.1 


30.6 


29.1 


27.6 


26.1 


24.6 


23.0 


21.5 


20.0 


1.5 


150 


38.1 


36.8 


35.5 


34.3 


33.0 


31.8 


30.5 


29.2 


28.0 


26.7 


25.5 


24.2 


23.0 


21.7 


1.3 


160 


36.5 


35.4 


34.4 


33.4 


32.4 


31.4 


30.4 


29.4 


28.4 


27.4 


26.4 


25.4 


24.4 


23.3 


1.0 


170 


34.8 


34.1 


33.3 


32.6 


31.8 


31.1 


30.3 


29.5 


28.8 


2S.0 


27.3 


26.5 


25.8 


25.0 


0.8 


180 


33.2 


32.7 


32.2 


31.7 


31.2 


30.7 


30.2 


29.7 


29.2 


28.7 


28.2 


27.7 


27.2 


26.7 


0.5 


190 


31.6 


31.4 


31.1 


30.9 


30.6 


30.4 


30.1 


29.8 


29.6 


29.3 


29.1 


28.8 


28.6 


28.3 


0.3 


200 


30.0 


30.0 


30.0 


30.0 


30.0 


30.0 


30.0 


30.0 


30.0 


80.0 


30.0 


30.0 


30.0 


30.0 


0.0 


210 


28.4 


28.6 


28.9 


29.1 


29.4 


29.6 


29.9 


30.2 


30.4 


30.7 


30.9 


31.2 


31.4 


31.7 


0.3 


220 


26.8 


27.3 


27.8 


28.3 


28.8 


29.3 


29.8 


30.3 


30.8 


31.3 


31.8 


32.3 


32.8 


33.3 


0.5 


230 


25.2 


25.9 


26.7 


27.4 


28.2 


28.9 


29.7 


30.5 


31.2 


32.0 


32.7 


33.5 


34.2 


35.0 


0.8 


240 


23.5 


24.6 


25.6 


26.6 


27.6 


28.6 


29.6 


30.6 


31.6 


32.6 


33.6 


34.6 


35.6 


36.7 


1.0 


250 21.9 


23.2 


24.5 


25.7 


27.0 


28.2 


29.5 


30.8 


32.0 


33.3 


34.5 


35.8 


37.1 


38.3 


1.3 


260 


20.3 


21.8 


23.3 


24.9 


26.4 


27.9 


29.4 


30.9 


32.4 


33.9 


35.4 


37.0 


38.5 


40.0 


1.5 


270 


18.7 


20.5 


22.2 


24.0 


25.8 


27.5 


29.3 


31.1 


32.8 


34,6 


36.3 


38.1 


39.9 


41.6 


1.8 


280 


17.1 


19.1 


21.1 


23.1 


25.2 


27.2 


29.2 


31.2 


33.2 


35.2 


37.3 


39.3 


41.3 


43.3 


2.0 


290 


15.5 


17.8 20.0 


22.3 


24.6 


26.8 


29.1 


31.4 


33.6 


35.9 


38.2 


40.4 


42.7 


45.0 


2.3 


300 


13.9 


16.4 18.9 


21.4 


24.0 


26.5 


29.0 


31.5 


34.0 


36.6 


39.1 


41.6 


44.1 


46.6 


2.5 


310 


12.3 


15.0 17.8 


20.6 


23.3 


26.1 


28.9 


31.7 


34.4 


37.2 


40.0 


42.7 


45.5 


48.3 


2.8 j 


320 


10.7 


13.7 


16.7 


19.7 


22.7 


25.8 


28.8 


31.8 


34.8 


37.9 


40.9 


43.9 


46.9 


50.0 


3.0 


330 


9.0 


12.3 


15.6 


18.9 


22.1 


25.4 


28.7 


32.0 


35.2 


38.5 


41.8 


45.1 


48.3 


51.6 


33 


340 


7.4 


10.9 


14.5 


18.0 


21.5 


25.1 


28.6 


32.1 


35.6 


39.2 


42.7 


46.2 


49.8 


53.3 


35 


350 


5.8 


9.6 


13.4 


17.1 


20.9 


24.7 


28.5 


32.3 36.0 


39.8 


43.6 


47.4 


51.2 


54.9 


3.8 


360 


4.2 


8.2 


12.3 


16.3 


20.3 


24.4 


28.4 


32.4 


36.4 


40.5 


44.5 


48.5 


52.6 


56.6 


4.0 


370 


2.6 


6.9 


11.1 


15.4 


19.7 


24.0 


28.3 32.6 


36.8 


41.1 


45.4 


49.7 


54.0 


58.3 


4 3 


380 


1.0 


5.5 


10.0 


14.6 


19.1 


23.6 


28.2 32.7 


37.2 


41.8 


46.3 


50.9 


55.4 


59.9 


4.5 


i 
i 


0" 


50" 


100" 


150" 


200" 


250" 


300" 350" 


400" 


450" 


500" 


550" 


600" 


650" 





TABLE LXXXI I. Moon's Horary Motion in Latitude. 
Argument. Arg. II. of Latitude. 



o 



Os 
9.3 


Is 

8.7 


Us 


His 


TVs 


Vs 


o 
30 


7.1 


5.0 


2.9 


1.3 


3 


9.3 


8.6 


6.9 


4.8 


2.7 


1.2 


27 


6 


9.2 


8.5 


6.7 


4.6 


2.5 


1.1 


24 


9 


9.2 


8.3 


6.5 


4.3 


2.3 


1.0 


21 


12 


9.2 


8.2 


6.3 


4.1 


2.1 


0.9 


18 


15 


9.1 


8.0 


6.1 


3.9 


2.0 


0.9 


15 


18 


9.1 


7.9 


5.9 


3.7 


1.8 


0.8 


12 


21 


9.0 


7.7 


5.7 


3.5 


1.7 


0.8 


9 


24 


8.9 


7.5 


5.4 


3.3 


1.5 


0.8 


6 


27 


88 


7.3 


5.2 


3.1 


1.4 


0.7 


3 


30 


87 
XL? 


7.1 


5.0 


2.9 


1.3 

Vlfs 


0.7 

Vis 





Xs 


ix» 


VULs 



M 



98 TABLE LXXXIII. 

Moon's Horary Motion in Latitude. 

Arguments. Preceding equation, and Sum 
of equations of Horary Motion in Longi- 
tude, except the last two. 



Preo. 




» 


" 


- 


" 


" 


- 


n 


equ. 





100 


200 


300 


400 


500 


600 


700 


// 





2.1 


1.8 


1.5 


1.2 


0.9 


0.6 


0.3 


0.0 


1 


1.9 


1.6 


1.4 


1.1 


0.9 


0.7 


0.4 


0.2 


2 


1.7 


1.5 


1.3 


1.1 


1.0 


0.8 


0.6 


0.3 


3 


1.5 


1.4 


1.2 


1.1 


1.0 


0.9 


0.8 


0.6 


4 


1.3 


1.2 


1.2 


1.1 


1.1 


1.0 


0.9 


0.9 


5 


1.1 


1.1 


1.1 


1.1 


1.1 


1.1 


1.1 


1.1 


6 


0.9 


1.0 


1.0 


1.1 


1.1 


1.2 


1.3 


1.3 


7 


0.7 


0.8 


1.0 


1.1 


1.2 


1.3 


1.4 


1.6 


8 


0.5 


0.7 


0.9 


1.1 


1.2 


1.4 


1.6 


1.9 


9 


0.3 


0.6 


0.8 


1.1 


1.3 


1.5 


1.8 


2.0 


10 


0.1 


0.4 


0.7 


1.0 


1.3 


1.6 


1.9 


2.2 




„ 


„ 


/, 


,, 


/• 







100 


200 


300 


400 


500 


600 


700 



Constant to be subtracted 237" .2. 

TABLE LXXXV. 

Moon's Horary Motion in Latitude. 
(Equations of second order.) 
Arguments. Preceding equation, and Sum 
of equations of Horary Motion in Longi- 
tude, except the last two. 



Prec. 


" 


" 


" 


" 


" 


" 


" 


, t 


equ. 





100 


200 


300 


400 


500 


600 


700 


0.00 0.65 


0.57 


0.48 


0.39 


0.31 


0.21 


0.12 


0.00 


0.10 0.62 0.55 


0.47 


0.39 


0.31 


0.23 


0.15 


0.04 


0.20 0.69 0.53 


0.46 


0.39 


0.32 


0.25 


0.18 


0.09 


0.30 0.66 0.51 


0.45 


0.39 


0.33 


0.27 


0.21 


0.13 


0.40 0.63 


0.48 


0.44 


0.39 


0.34 


0.29 


0.24 


0.17 


0.50 0.50 


0.46 


0.43 


0.38 


0.35 


0.30 


0.27 


0.21 


0.60 0.47 


0.44 


0.42 


0.38 


0.36 


0.32 


0.29 


0.25 


0.70 0.44 


0.42 


0.40 


0.38 


0.36 


0.34 


0.32 


0.30 


0.80 0.41 


0.40 


0.39 


0.38 


0.37 


0.36 


0'.35 


0.34 


0.90 


0.38 


0.38 


0.38 


0.38 


0.38 


0.38 


0.38 


0.38 


1.00 


0.35 


0.36 


0.37 


0.38 


0.39 


0.40 


0.41 


0.42 


1.10 


0.32 


0.34 


0.36 


0.38 


0.40 


0.42 


0.44 


0.46 


1.20 


0.29 


0.32 


0.34 


0.38 


0.40 


0.44 


0.47 


0.51 


1.30 


0.26 


0.30 


0.33 


0.38 


0.41 


0.46 


0.49 


0.55 


1.40 


0.23 


0.28 


0.32 


0.37 


0.42 


0.47 


0.52 


0.59 


1.50 


0.20 


0.25 


0.31 


0.37 


0.43 


0.49 


0.55 


0.63 


1.60 


0.17 


0.23 


0.30 


0.37 


0.44 


0.51 


0.58 


0.67 


1.70 


0.14 


0.21 


0.29 


0.37 


0.45 


0.53 


0.61 


0.72 


1.80 


0.11 



0.19 
100 


0.28 
200 


0.37 
300 


0.45 
400 


0.55 


0.04 


0.76 
700 


500 


600 



TABLE LXXXIV. 

Moon's Hor. Motion in Lat 

(Equa. of second order.) 

Argument. Arg. I of Lat. 





I 


I 




s ° 


" 


» 


s ° 


O 


0.90 


0.90 


XII 


5 


0.83 


0.97 


25 


10 


0.75 


1.05 


20 


15 


0.68 


1.12 


15 


20 


0.61 


1.19 


10 


25 


0.54! 


1.26 


5 


I 


0.47 


1.33 


XI 


5 


0.41 


1.39 


25 


10 


0.351 


1.45 


20 


15 


0.29! 


1.51 


15 


20 


0.24! 


1.56 


10 


25 


0.20 ! 


1.60 


5 


II 


0.16 


1.64 


X 


5 


0.12 


1.68 


25 


10 


0.09! 


1.71 


20 


15 


0.07j 


1.73 


15 


20 


0.05 


1.75 


10 


25 


0.041 


1.76 


5 


III 


0.04 


1.76 


IX 


5 0.04 


1.76 


25 


10 0.05 


1.75 


20 


15 0.07 1 


1.73 


15 


20 0.09! 


1.71 


10 


25 0.12 1 


1.08 


5 


IV 0.161 


1.64 


VIII 


5 0.20 


1.60 


25 


10 ,0.24' 


1.56 


20 


15 ( 0.29 


1.51 


15 


20 0.35 


1.45 


10 


25 .0.41 


1.39 


5 


V 0.47 


1.33 


VII 


5 0.54 


1.26 


25 


10 0.61 


1.19 


20 


15 0.68 


1.12 


15 


20 0.751 


1.05 


10 


25 0.83 


0.97 


5 


VI 0.901 


0.90 


VI 



TABLE LXXXVI. 

Mean New Moons and Arguments, in January. 



99 





Mean New 












Years. 


Moon in. 
January. 


I. 


II. 


III. 


IV. 


N. 




d. h m. 












1821 


2 17 59 


0092 


7859 


80 


78 


823 


1S22 


21 15 32 


0602 


7182 


78 


66 


930 


1823 


11 20 


0304 


5787 


61 


55 


953 


1824 B 


29 21 53 


0814 


5110 


59 


43 


060 


1825 


18 6 41 


0516 


3716 


42 


32 


083 


1826 


7 15 30 


0218 


2321 


25 


21 


105 


1S27 


26 13 3 


0728 


1644 


24 


09 


213 


1S28B 


15 21 51 


0430 


0250 


07 


98 


235 


1S29 


4 6 40 


0131 8855 


90 


87 


257 


1830 


23 4 12 


0642 


8178 


88 


75 


365 


1831 


12 13 1 


0343 


67S4 


71 


64 


387 


1832 B 


1 21 50 


0045 


5389 


54 


53 


409 


1833 


19 19 22 


0555 


4712 


53 


42 


517 


1834 


9 4 11 


0257 


3318 


36 


31 


539 


1835 


28 1 43 


0768 


2641 


34 


19 


647 


1836 B 


17 10 32 


0469 


1246 


17 


08 


669 


1837 


5 19 20 


01/1 


9852 


00 


97 


692 


1838 


24 16 53 


0681 9175 


99 


85 


799 


1839 


14 1 42 


0383 


7780 


82 


74 


822 


1840 B 


3 10 30 


0085 


6386 


65 


63 


844 


1841 


21 8 3 


0595 


5709 


63 


51 


951 


1842 


10 16 51 


0.297 


4314 


46 


40 


974 


1843 


29 14 24 


0S07 


3637 


44 


28 


081 


1844 B 


18 23 13 


0509 


2243 


28 


17 


104 


1845 


7 8 1 


0211 


084S 


11 


06 


126 


1846 


26 5 34 


0721 


0171 


09 


94 


234 


1847 


15 14 22 


0423 


8777 


92 


84 


256 


i848 B 


4 23 11 


0125 


7382 


75 


73 


278 


1849 


22 20 43 


0635 


6705 


73 


61 


386 


1850 


12 5 32 


0337 


5311 


56 


50 


408 


1851 


1 14 21 


0038 


3916 


40 


39 


431 


1852 B 


20 11 53 


0549 


3239 


38 


27 


538 


1853 


8 20 42 


0251 


1845 


21 


16 


560 


1854 


27 18 14 


0761 


1168 


19 


04 


668 


1855 


17 3 3 


0463 i 


9773 


02 


93 


690 


1856 B 


6 11 51 


0164 ! 


8379 


85 


82 


713 


1857 


24 9 24 


0675; 


7702! 


84 


70 


820 


1853 


13 18 13 


0376 


6307! 


67 


59 


S43 


1859 


3 3 1 


0078 ! 4913 j 50 


48 


865 


1860 B 


22 34 


0588 | 4236 1 48 


36 


972 



:oo 



TABLE LXXXVII. 



Mean Lunations and Changes of the Arguments. 



Num 


Lunations. 


I. 


II. 


III. 


IV. 


N. 




d. h m 












1 
2 


14 18 22 


404 


5359 


58 


50 


43 


1 


29 12 44 


808 


717 


15 


99 


85 


2 


59 1 28 


1617 


1434 


31 


98 


170 


3 


88 14 12 


2425 


2151 


46 


97 


256 


4 


118 2 56 


3234 


2869 


61 


96 


341 


5 


147 15 40 


4042 


3586 


76 


95 


426 


6 


177 4 24 


4851 


4303 


92 


95 


511 


7 


206 17 8 


5659 


5020 


7 


94 


596 


8 


236 5 52 


6468 


5737 


22 


93 


682 


9 


265 18 36 


7276 


6454 


37 


92 


767 


10 


295 7 20 


8085 


7171 


53 


91 


852 


11 


324 20 5 


8893 


7889 


68 


90 


937 


12 


354 8 49 


9702 


8606 


83 


89 


22 


13 


383 21 33 


510 


9323 


98 


88 


108 

- j 



TABLE LXXXVIII. 

Number of Days from the commencement of the year 
to the first of each month 



Months. 


Com. 


Bis. 


January 
February 
March . 








Days. 

31 
59 


Days. 

31 
60 


April . 

May 

June 








90 
120 
151 


91 
121 

152 


JuJy . 
August . 
September 
October 








181 
212 
243 
273 


182 
213 
244 
274 


November 






304 


305 


December 






334 


335 



TABLE LXXXIX. 

Equations for New and Full Moon. 



101 



Arg. 


I 


II 


Arg. 


I 


II 


Arg 


™ 


H. 


Arg 


h m 


h m 




h m 


h m 




m 


m 







4 20 


10 10 


5000 


4 20 


10 10 


25 


3 


31 


25 


100 


4 36 


9 36 


5100 


4 5 


10 50 


26 


3 


31 


24 


200 


4 52 


9 2 


5200 


3 49 


11 30 


27 


3 


30 


23 


300 


5 8 


8 28 


5300 


3 34 


12 9 


28 


3 


30 


22 


400 


5 24 


7 55 


5400 


3 19 


12 48 


29 


3 


30 


21 


500 


5 40 


7 ^2 


5500 


3 4 


13 26 


30 


3 


30 


20 


600 


5 55 


6 49 


5600 


2 49 


14 3 


31 


3 


30 


19 


700 


6 10 


6 17 


5700 


2 35 


14 39 


32 


4 


30 


18 


800 


6 24 


5 46 


5800 


2 21 


15 13 


|33 


4 


29 


17 


900 


6 38 


5 15 


5900 


2 8 


15 46 


34 


4 


29 


16 


1000 


6 51 


4 46 


6000 


1 55 


16 18 


35 


4 


29 


15 


1100 


7 4 


4 17 


6100 


1 42 


16 48 


36 


5 


2S 


14 


1200 


7 15 


3 50 


6200 


1 31 


17 16 


37 


5 


28 


13 


1300 


7 27 


3 24 


6300 


1 19 


17 42 


38 


5 


27 


12 


j 1400 


7 37 


2 59 


6400 


1 9 


18 6 


39 


5 


27 


11 


1500 


7 47 


2 35 


6500 


59 


18 28 


40 


6 


26 


10 


1600 


7 55 


2 14 


6600 


50 


18 48 


41 


6 


26 


9 


1700 


8 3 


1 53 


6700 


42 


19 6 


42 


7 


25 


8 


1800 


8 10 


1 35 


6800 


34 


19 21 


43 


7 


25 


7 


1900 


8 16 


1 18 


6900 


28 


19 33 


44 


7 


24 


6 


2000 


8 21 


1 3 


7000 


22 


19 44 


45 


8 


23 


5 


2100 


8 25 


51 


7100 


17 


19 52 


46 


S 


23 


4 


2200 


8 29 


40 


7200 


14 


19 57 


47 


9 


22 


3 


2300 


8 31 


32 


7300 


11 


20 


48 


9 


21 


2 


2400 


8 32 


25 


7400 


9 


20 1 


49 


10 


21 


1 


2500 


8 32 


21 


7500 


8 


19 59 


50 


10 


20 





2600 


8 31 


19 


7600 


8 


19 55 


51 


10 


19 


99 


2700 


8 29 


20 


7700 


9 


19 48 


52 


11 


19 


98 


2800 


8 26 


23 


7800 


11 


19 40 


53 


11 


18 


97 


2900 


8 23 


28 


7900 


15 


19 29 


54 


12 


17 


96 


3000 


8 18 


36 


8000 


19 


19 17 


55 


12 


17 


95 


3100 


8 12 


47 


8100 


24 


19 2 


56 


13 


16 


94 


3200 


8 6 


59 


8200 


30 


18 45 


57 


13 


15 


93 


3300 


7 58 


1 14 


8300 


37 


18 27 


58 


13 


15 


92 


3400 


7 50 


1 32 


8400 


45 


18 6 


59 


14 


14 


91 


3500 


7 41 


1 52 


8500 


53 


17 45 


60 


14 


14 


90 


3600 


7 31 


2 14 


8600 


1 3 


17 21 


61 


15 


13 


89 


3700 


7 21 


2 38 


8700 


1 13 


16 56 


62 


15 


13 


88 


3800 


7 9 


3 4 


8800 


1 25 


16 30 


63 


15 


12 


87 


3900 


6 58 


3 32 


8900 


1 36 


16 3 


64 


15 


12 


86 


4000 


6 45 


4 2 


9000 


1 49 


15 34 


65 


16 


11 


85 


4100 


6 32 


4 34 


9100 


2 2 


15 5 


66 


16 


11 


84 


4200 


6 19 


5 7 


9200 


2 16 


14 34 


67 


16 


11 


83 


4300 


6 5 


5 41 


9300 


2 30 


14 3 


68 


1G 


10 


82 


4400 


5 51 


6 17 


9400 


2 45 


13 31 


69 


17 


10 


81 


4500 


5 36 


6 54 


9500 


3 


12 58 


70 


17 


10 


80 


4600 


5 21 


7 32 


9600 


3 16 


12 25 


71 


17 


10 


79 


4700 


5 6 


8 11 


9700 


3 32 


11 52 


72 


17 


10 


78 


4800 


4 51 


8 50 


9800 


3 48 


11 18 


73 


17 


10 


77 


4900 


4 35 


9 30 


9900 


4 4 


10 44 


74 


17 


9 


76 


5000 


4 20 


10 10 


10000 


4 20 


10 101 


75 


17 


9 


75 



102 



TAELE XC. 



Mean Right Ascensions and Declinations of 50 principal Fixed 
Stars, for the beginning of 1 840. 



Stars' Name. 


Mag 


Right Ascen. 


AnnualVar. 


Declination. 


Ann. Var. ! 


1 Algenib 

2 ,8 Andromedae 
| 3 Polaris 

\ 4 Achernar 
5 aArietis 


2.3 
2 
2.3 

1 
3 


h m s 

5 0.31 

1 46.7 

1 2 10.38 
1 31 44.88 
1 58 9.94 


+ 3.0775 

3.309 

16.1962 

2.2351 

3.3457 


' " 
14 17 38.82 N 
34 46 17.2 N 
88 27 21.96 N 
58 3 5.13 S 
22 42 11.81 N 


-f 20.051 1 

19.35 ; 

19.339 | 

— 18.473 

+ 17.455 ; 


6 a Ceti 

7 a Persei 

8 Aldebaran 

9 Capella 
10 2tt#eZ 


2.3 
2.3 

1 

1 
1 


2 53 55.34 

3 12 55.97 

4 26 44.77 

5 4 52.67 
5 6 51.09 


+ 3.1257 
4.2280 
3.4264 
4.4066 
2.8783 


3 27 30.09 N 

49 17 8.74N 

16 10 56.82 N 

45 49 42.81 N 

8 23 29.29 S 


+ 14.561 ! 

13.371 i 

7.949 1 

4.793 j 

— 4.620 


11 BTami 

12 y Orionis 

13 a Columbao 

14 a Orionis 

1 5 Canopus 


2 
2 

2 
1 
1 


5 16 10.96 
5 16 33.1 
5 33 51.52 

5 46 30.71 

6 20 24.18 


+ 3.7820 
3.210 
2 16S8 
3.2430 
1.3278 


28 27 58.20 N 

6 11 55.3 N 
34 9 47.41 S 

7 22 17.14N 
52 36 38.42 S 


+ 3.S25 
+ 3.82 
— 2.291 
+ 1.191 
1.77S 


1 6 Sirius 

17 Castor 

18 Procyon 

19 PoHmz 

20 aHvdrae 


1 
3 
1.2 

2 
2 


6 38 5.76 

7 24 23.06 
7 30 55.53 
7 35 31.07 
9 19 43.57 


+ 2.6458 
3.8572 
3.1448 
3.6840 
2.9500 


16 30 4.79 S 
32 13 58.89 N 

5 37 48.92 N 
28 24 25.57 N 

7 58 4.83 S 


+ 4.449 

— 7.206 

8.720 

8.107 

+ 15.341 i 


21 Regulus 

22 a Ursae Majoris 

23 /?Leonis 

24 /?Virginis 

25 y Ursae Majoris 


1 

1.2 
2.3 
3.4 

2 


9 59 50.93 

10 53 47.98 

11 40 53.69 
11 42 21.4 
11 45'22.93 


-f 3.2220 
3.8077 
3.0660 
3.124 
3.1914 


12 44 49.70 N 
62 36 48.93 N 
15 28 1.16N 

2 40 2 6 N 
54 35 4 67N 


— 17.356 

19.221 
19.985 
19.98 
20.014 


26 a 2 Crucis 

27 Spica 

28 6 Centauri 

29 a Draconis 

30 Arcturus 


2 

1 
2 
3.4 

1 


12 17 43.7 

13 16 46.36 

13 57 18.0 

14 2.8 
14 8 21.96 


+ 3.258 
3.1502 
3.491 
1.625 
2.7335 


62 12 47. 9S 
10 19 24.39 S 
35 34 41.9 S 
65 8 32.1 N 
20 1 7.67 N 


+ 19.99 
18.945 
17.499 

— 17.37 
18.956 


31 a 2 Centauri 

32 a 2 Librae 

33 (o Ursae Minoris 

34 y 2 Ursae Minoris 

35 a Coronae Borealis 


1 
3 
3 
3.4 

2 


14 28 47.84 
14 42 2.44 

14 51 14.66 

15 21 1.3 
15 27 54.87 


+ 4.00S6 
3.3088 

— 0.2787 

— 0.179 
+ 2.5277 


60 10 6.24 S 
15 22 18.25 S 
74 48 34.1S N 
72 24 14.1 N 
27 15 27.71 N 


+ 15.152 
15.256 

— 14.712 
12.81 
12.361' 


36 a Serpentis 

37 /?Scorpii 

38 Antares 

39 a Herculis 

40 a Ophiuchi 


2.3 

2 
1 
3.4 

2 


15 36 23.43 

15 56 8.68 

16 19 36.49 

17 7 21.30 
17 27 30.56 


+ 2.93S6 
3.4729 
3.6625 
2.7317 
2.7724 


6 56 2.80 N 
19 21 3S.82S 
26 4 13.13 S 
14 34 41.43 N 
12 40 58.65 N 


— 11.770 
+ 10.330 

8.519 

— 4.576 
2.844 


41 6 Ursae Minoris 

42 Vega 

43 Altair 

44 a 2 Capricorni 

45 a Cygni 


3 

1 

1 
3 

1 


18 23 56.48 

18 31 31.19 

19 42 58.61 

20 9 10.34 
20 35 58.80 


— 19.2072 

+ 2.0116 

2.9255 

3.3323 

2.0416 


86 35 28.89 N 
38 38 16.85N 
8 27 0.21 N 
13 2 5.57 S 
44 42 41.38 N 


+ 2.161 

2.742 

8.701 

— 10.705 

+ 12.614 


46 a Aquarii 

47 Fomalkaut 

48 /JPegasi 

49 Markab 

50 a Andromedae 


3 

1 
2 
2 
1 


21 57 33.93 

22 48 47.67 
22 56 1.1 
22 56 47.75 
24 7.72 


+ 3.0835 
3.3114 
2.878 
2.9771 
3.0704 


1 5 3S.00S 
30 28 4.91 S 

27 13 1.7 N 
14 20 46.92 N 

28 12 27.00 N 


— 17.256 

19.092 

+ 19.255 

19.295 

20.056 



TABLE XCI. 



Constants for the 


Aberration and Nutation in it 


j'ght Ascension 


and Declination of the Stars in the preceding Catalogue 


i 


Aberration. 


Nutation. 


s ° ' 


M | 


» i 


N 


V 


M' 


6> 


N' 




s ° ' 




s ° ' 




s o > 




l 


8 28 47 


0.1087 


7 27 12 


0.9657 


6 8 24 


0.0300 


5 28 30 


0.8381 


2 


8 13 39 


0.1830 


6 19 12 


1.0740 


6 19 53 


0.0838 


5 10 8 


0.8496 


3 


8 13 51 


1.6526 


5 16 57 


1.3052 


8 16 7 


1.3427 


5 10 22 


0.8493 


4 


8 5 20,0.3801 


10 26 46 


1.2798 


4 10 12 


0.0775 


5 31 


0.8629 


5 


7 28 26 


0.1397 


7 2 


0.8972 


6 11 1 


0.0695 


4 22 53 


0.8765 


6 


7 14 11 


0.1149 


8 23 8 


0.8678 


6 1 26 


0.0322 


4 8 16 


0.9078 


7 


7 9 30 


0.3020 


5 3 5 


1.0630 


6 18 13 


0.1849 


4 3 47 


0.9179 


8 


6 21 43 


0.1447 


7 23 12 


0.5760 


6 3 27 


0.0726 


3 17 54 


0.9502 


91 


6 12 51 


0.2S75 


3 25 37 


0.9112 


6 5 46 


0.1830 


3 10 29 


0.9605 


io ! 


6 12 20 


0.1355 


9 3 42 


1.0300 


5 28 47 


1.9966 


3 10 4 


0.9608 


11 


6 10 13 


0.1873 


4 19 21 


0.3917 


6 2 52 


0.1008 


3 8 19 


0.9626 


12 


6 10 6 


0.1340 


8 25 4 


0.7851 


6 40 0.0441 


3 8 14 


0.9626 


13 


6 6 5 


0.2145 


9 4 24 


1.2348 


5 26 18 


1.8750 


3 4 57 


0.9648 


14 


6 3 13 


0.1361 


8 28 23 


0.7521 


6 15 


0.0481 


3 2 37 


0.9657 


15 


5 25 22 


0.3491 


8 25 53 


1.2960 


6 8 46 


1.6679 


2 26 15 


0.9657 


16 


5 21 21 


0.1501 


8 25 51 


1.1152 


6 1 51 


1.9658 


2 22 58' 


0.9636 


17 


5 10 40 


0.2010 


1 2 17 


0.6620 


5 24 2 


0.1257 


2 14 6 


0.9535 


18 


5 9 6 


0.1297 


9 6 54 


0.8071 


5 28 47 


0.0414 


2 12 47 


0.9513 


19 


5 8 2 


0.1829 


14 32 


0.6052 


5 24 2 


0.1114 


2 11 53 


0.9499 


20 


4 12 39 


0.1158 


8 17 31 


0.9967 


6 3 41 


0.0081 


1 18 37 


0.9007 


21 


4 2 22 


0.1162' 


10 3 47 


0.8457 


5 23 47 


0.0480 


1 7 59 


0.8782 


22 


3 18 7 


0.4366 


3 28 


1.2394 


4 18 58 


0.2407 


21 57 


0.8520 


23 


3 5 21 


0.1117 


10 6 20 


0.9621 


5 20 56 


0.0344 


6 35 


0.8393 


24 


3 4 57 


0.0958 


9 6 51 


0.9075 


5 28 25 


0.0253 


6 5 


0.8390 


25 


3 4 8 


0.3229 


11 17 28 


1.2298 


4 21 46 


0.1465 


5 5 


0.8388 | 


26 


2 25 19 


0.4261 ' 


6 8 5 


1.2585 


7 16 2 


0.2089 


11 24 14 


0.8390 


27 


2 9 22 


0.1066 


8 3 31 


0.8862 


6 5 51 


0.0154 


11 5 6 


0.8559 


28 


1 28 40 


0.1942 


6 7 12 


1.0176 


6 17 31 


0.1062 


10 23 8 


0.8760 


29 


1 27 53 


0.4824 ! 


10 23 28 


1.2995 


3 25 50 


0.1090 


10 22 16 


0.8777 


30 


1 25 46 


0.1336 


9 28 18 


1.0974 


5 18 49 


1.9937 


10 20 1 


0.8822 


31 


1 20 32 


0.4123 


5 7 54 


1.1820 


6 29 6 


0.2460 


10 14 36 


0.8937 


32 


1 17 26 


0.1273 


7 18 24 


0.6923 


6 6 29 


0.0593 


10 11 28 


0.9006 i 


33 


i 14 42 


0.6961 


10 15 5 


1.3087 


2 26 45 


0.2235 


10 8 47 


0.9066! 


34 


1 7 20 


0.63S6 


10 7 33 


1.3087 


2 27 7 


0.0960 


10 1 45 


0.9225 j 


35 


1 5 45 


0.1704 

i 


9 22 28 


1.1785 


5 17 18 


1.9510 


10 18 


0.92571 


36 


1 3 43 0.1237 


9 8 22 


0.9994 


5 27 30 


0.0058 


9 28 26 


0.9298 ! 


37 


28 58 0.1485 


7 4 4 


0.6237 


6 5.20 


0.0795 


9 24 12 


0.9386 


38 


23 24 0.1723 


5 27 59 


0.5816 


6 5 49 


0.1029 


9 19 21 


0.9478 


39 


12 13 0.1451 


9 5 25 


1.0962 


5 27 45 


1.9742 


9 9 58 


0.9610 


40 


7 34 0.1427 


9 3 4 


1.0786 


5 28 48 


1.9803 


9 6 9 


0.9642 


41 


11 23 47 


1.3571 


8 22 49 


1.2821 


11 19 31 


0.8257 


8 24 57 


0.9650 


42 


11 22 50 


0.2393 


8 24 29 


1.2545 


6 5 31 


1.8436 


8 24 10 


0.9644 


43 


11 6 15 


0.1309 


8 22 59 


1.0237 


6 2 16 


1.9988 


8 10 21 


0.9472 


44 


11 2 0.1341 


9 29 33 


0.6961 


5 26 12 


0.06J9 


8 4 55 


0.9368 


45 


10 23 29 


0.2668 


| 8 39 


1.2634 


6 28 32 


1.9042 


7 29 


0.9242 


46 


10 2 57 


0.1057 


' 9 2 31 


0.8988 


5 29 26 


0.0264 


7 8 37 


0.8794 


47 


9 19 26 


! 0.1638 


11 7 34 


1.0271 


5 13 8 


0.0765 


6 23 30 


0.8540 


48 


9 17 29 


i 0.1491 


7 17 


1.1171 


6 17 2 


0.0162 


6 21 13 


0.8511 


49 


9 17 17 


! 0.1120 


8 2 5 


1.0138 


6 8 23 


0.0157 


6 20 58 


0.8508 


150 


9 6 0.1495 


7 6 42 


1.0785 


6 17 20 


0.0444 


6 8 


0.8380 



104 



TABLE XCII. 



Mean Longitudes and Latitudes of some of the principal Fixed 
Stars for the beginning of 1840, with their Annual Variations. 



Stars' Name. 


Mag 


Longitude. 


Annual 
Var. 


Latitude. 


Annual 
Var. 


a Arietis 


3 


1 5 25 


27.6 


50.277 


O r " 

9 57 40.9 N 


+ 0.161 


Aldebaran 


1 


2 7 33 


5.9 


50.210 


5 28 38.0 S 


— 0.335 


Capella 


1 


2 19 37 


17.8 


50.302 


22 51 44.4 N 


— 0.052 


Polaris 


2.3 


2 26 19 


20.1 


47.959 


66 4 59.5 N 


+ 0.552 


Sirius 


1 


3 11 52 


32.9 


49.48S 


39 34 4.3 S 


+ 0.319 


Canopus 


1 


3 12 44 


59.6 


49.366 


75 50 57.6 S 


4 0.459 


Pollux 


2 


3 21 


22.0 


49.502 


6 40 20.2 N 


-t- 0.255 


Regulus 


1 


4 27 36 


13.2 


49.946 


27 38.3 N 


+ 0.220 


Spica 


1 


6 21 36 


29.2 


50.085 


2 2 29.7 S 


+ 0.171 


Arcturus 


1 


6 22 


4.7 


50.711 


30 51 17.5 N 


+ 0.214 


Antares 


1 


8 7 31 


45.2 


50.120 


4 32 51.6 S 


4 0.424 


Altair 


1.2 


9 29 31 


5.9 


50.795 


29 18 37.3 N 


4 0.080 


Fomalhaut 


1 


11 1 36 


22.0 


50.595 


21 6 49.7 S 


4 0.213 


Achernar 


1 


11 13 2 


5.3 


50.346 


17 6 17.3 S 


— 0.083 


a Pegasi 


2 


11 21 15 


24.7 


50.112 


19 24 40.9 N 


4 0.098 



TABLE added to TABLE XC. 

Mean Right Ascensions and Declinations of Polaris and s Ursae 
Minoris for 1830, 1840, 1850, and 1860. 



Stars. 


Years 


Right Asc. 


Ann. Var. 


Declination. 


Ann. Var. 


Polaris 
i Ursae Minoris 


1830 
1840 
1850 
1860 

1830 
1840 
1850 
1860 


O » '- 

59 30.76 

1 2 10.32 
1 5 0.29 
1 8 1.79 

18 27 5.13 
18 23 53.03 
18 20 40.21 
18 17 26.77 


+ 15.478 
16.470 
17.567 
18.784 

— 19.167 
19.241 
19.305 
19.360 


O / " 

88 24 8.82 
88 27 22.43 
88 30 35.40 
88 33 47.64 

86 35 5.70 
86 35 27.93 
86 35 47.36 
86 36 3.97 


// 

4 19.371 

19.309 

19.240 

19.163 

4 2.363 
2.085 
1.805 
1.523 



TABLE XCIII. 
Second Differences. 



105 



Hours & Minutes. 


r 


2' 


3' 


• 4' 


5' 


6' 


r 


8' 


9' 


io- 


ir 


h m 


h m 


" 


« 


" 


>> 


ir 


" 


ir 


/- 


f> 


" 


" 





12 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 


10 


11 50 


0.4 


0.8 


1.2 


1.6 


2.0 


2.4 


2.9 


3.3 


3.7 


4.1 


4.5 


20 


11 40 


0.8 


1.6 


2.4 


3.2 


4.1 


4.9 


5.7 


6.5 


7.3 


81 


8.9 


30 


11 30 


1.2 


2.4 


3.6 


4.8 


6.0 


7.2 


8.4 


9.6 


10.8 


12.0 


]3.2 


40 


11 20 


1.6 


3.1 


4.7 


6.3 


7.9 


9.4 


11.0 


12.6 


14.2 


15.7 


17.3 


50 


11 10 


1.9 


3.9 


5.8 


7.8 


9.7 


11.6 


13.6 


15.5 


17.4 


19.4 


21.4 


1 


11 


2.3 


4.6 


6.9 


9.2 


11.5 


13.8 


16.0 


18.3 


20.6 


22.9 


25.2 


1 10 


10 50 


2.6 


5.3 


7.9 


10.5 


13.2 


15.8 


18.4 


21.1 


23.7 


26.3 


29.0 


1 20 


10 40 


3.0 


5.9 


8.9 


11.9 


14.8 


17.8 


20.7 


23.7 


26.7 


29.6 


32.6 


1 30 


10 30 


3.3 


6.6 


9.8 


13.1 


16.4 


19.7 


23.0 


26.3 


29.5 


32.8 


36.1 


1 40 


10 20 


3.6 


7.2 


10.8 


14.4 


17.9 


21.5 


25.1 


28.7 


32.3 


35.9 


39.5 


1 50 


10 10 


3.9 


7.8 


11.6 


15.5 


19.4 


23.3 


27.2 


31.0 


34.9 


38.8 


42.7 


2 


10 


4,2 


8.3 


12.5 


16.7 


20.8 


25.0 


29.2 


33.3 


37.5 


41.7 


45.8 


2 10 


9 50 


4.4 


8.9 


13.3 


17.8 


22.2 


26.6 


31.1 


35.5 


40.0 


44.4 


48.8 


2 20 


9 40 


4.7 


9.4 


14.1 


18.8 


23.5 


28.2 


32.9 


37.6 


42.3 


47.0 


51.7 


2 30 


9 30 


4.9 


9.9 


14.8 


19.8 


24.7 


29.7 


34.6 


39.6 


44.5 


49.5 


54.4 


2 40 


9 20 


5.2 


10.4 


15.6 


20.7 


25.9 


31.1 


36.3 


41.5 


46.7 


51.9 


57.0 


2 50 


9 10 


5.4 


10.8 


16.2 


21.6 


27.1 


32.5 


37.9 


43.3 


48.7 


54,1 


59.5 


3 


9 


5.6 


11.3 


16.9 


22.5 


28.1 


33.8 


39.4 


45.0 


50.6 


56.3 


61.9 


3 10 


8 50 


5.8 


11.7 


17.5 


23.3 


29.1 


35.0 


40.8 


46.6 


52.4 


58.3 


64.1 


3 20 


8 40 


6.0 


12.0 


18.1 


24.1 


30.1 


36.1 


42.1 


48.1 


54.2 


60.2 


66.2 


3 30 


8 30 


6.2 


12.4 


18.6 


24.8 


31.0 


37.2 


43.4 


49.6 


55.8 


62.0 


68.2 


3 40 


8 20 


6.4 


12.7 


19.1 


25.5 


31.8 


38.2 


44.6 


50.9 


57.3 


63.7 


70.0 


3 50 


8 10 


6.5 


13.0 


19.6 


26.1 


32.6 


39.1 


45.7 


52.2 


58.7 


65.2 


71.7 


4 


8 


6.7 


13.3 


20.0 


26.7 


33.3 


40.0 


46.7 


53.3 


60.0 


66.7 


73.3 


4 10 


7 50 


6.8 


13.6 


20.4 


27.2 


34.0 


40.8 


47.6 


54.4 


61.2 


68.0 


74.8 


4 20 


7 40 


6.9 


13.8 


20.8 


27.7 


34.6 


41.5 


48.4 


55.4 


62.3 


69.2 


76.1 


4 30 


7 30 


7.0 


14.1 


21.1 


28.1 


35.2 


42.2 


49.2 


56.2 


63.3 


70.3 


77.3 


4 40 


7 20 


7.1 


14.3 


21.4 


28.5 


35.6 


42.8 


49.9 


57.0 


64.2 


71.3 


78.4 


4 50 


7 10 


7.2 


14.4 


21.6 


28.9 


36.1 


43.3 


50.5 


57.7 


64.9 


72.2 


79.4 


5 


7 


7.3 


14.6 


21.9 


29.2 


36.5 


43.8 


51.0 


58.3 


65.6 


72.9 


80.2 


5 10 


6 50 


7.4 


14.7 


22.1 


29.4 


36.8 


44.1 


51.5 


58.8 


66.2 


73.6 


80.9 


5 20 


6 40 


7.4 


14.8 


22.2 


29.6 


37.0 


44.4 


51.9 


59.3 


66.7 


74.1 


81.5 


5 30 


6 30 


7.4 


14.9 


22.3 


29.8 


37.2 


44.7 


52.1 


59.6 


67.0 


74.5 


81.9 


5 40 


6 20 


7.5 


15.0 


22.4 


29.9 


37.4 


44.9 


52.3 


59.8 67.3 


74.8 


82.2 


5 50 


6 10 


7.5 


15.0 


22.5 


30.0 


37.5 


45.0 


52.5 


60.0 67.4 


74.9 


82.4 


6 


6 

_ 


7.5 


15.0 


22.5 


30.0 


37.5 


45.0 


52.5 


60.0 67.5 


75.0 


82.5 j 



106 



TABLE XCIII. 

Second Differences. 



Hours & Min. j 


10" 


20" 


30" 


40" 


60 j 


1" 


2" 


3" 


4" 


5" 


6" 


7" 


8" | 


9" 


h m 


h m 


" 


~7> 


"77 


" 


" 


" 


"77 


"" 


"77 


>> 


"77 


~7 


"77777 





12 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 ' 0.0 


10 


11 50 


0.1 


0.1 


0.2 


0.3 


0.3 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 


0.1 


0.1 


20 


11 40 


0.1 


0.3 


0.4 


0.5 


0.7 


0.0 


0.0 


0.0 


0.1 


0.1 


0.1 


0.1 


0.1 


0.1 


30 


11 30 


0.2 0.4 


0.6 


0.8 


1.0 


0.0 


0.0 


0.1 


0.1 


o.i 


0.1 


0.1 


0.2 


0.2 


40 


11 20 0.3 


0.5 


0.8 


1.0 


1.3 


0.0 


0.1 


0.1 


0.1 


0.1 


0.2 


0.2 


0.2 


0.2 


50 


11 10 


0.3 


0.6 


1.0 


1.3 


1.6 


0.0 


0.1 


0.1 


0.1 


0.2 


0.2 


0.2 


0.3 


0.3 


1 


11 


0.4 


0.8 


1.1 


1.5 


1.9 


0.0 


0.1 


0.1 


0.2 


0.2 


0.2 


0.3 


0.3 


0.3 


1 10 


10 50 


0.4 


0.9 1.3 


1.8 


2.2 


0.0 


0.1 


0.1 


0.2 


0.2 


0.3 


0.3 


0.4 


0.4 


1 20 


10 40 


0.5 


1.0 


1.5 


2.0 


2.5 


0.0 


0.1 


0.1 


0.2 


0.2 


0.3 


0.3 


0.4 


0.4 i 


1 30 


10 30 


0.5 


1.1 


1.6 


2.2 


2.7 


0.1 


0.1 


0.2 


0.2 


0.3 


0.3 


0.4 


0.4 


0.5 j 


1 40 


10 20 


0.6 


1.2 


1.8 


2.4 


3.0 


0.1 


0.1 


0.2 


0.2 


0.3 


0.4 


0.4 


0.5 


0.5 


1 50 


10 10 


0.6 


1.3 


1.9 


2.6 


3.2 


0.1 


0.1 


0.2 


0.3 


0.3 


0.4 


0.5 


0.5 


0.6 


2 


10 


0.7 


1.4 


2.1 


2.8 


3.5 


0.1 


0.1 


0.2 


0.3 


0.3 


0.4 


0.5 


0.6 


0.6 


2 10 


9 50 


0.7 


1.5 


2.2 


3.0 


3.7 


0.1 


0.1 


0.2 


0.3 


0.4 


0.4 


0.5 


0.6 


0.7 


2 20 


9 40 


0.8 


1.6 


2.3 


3.1 


3.9 


0.1 


0.2 


0.2 


0.3 


0.4 


0.5 


0.5 


0.6 


0.7 


2 30 


9 30 


0.8 


1.6 


2.5 


3.3 


4.1 


0.1 


0.2 


0.2 


0.3 


0.4 


0.5 


0.6 


0.7 


0.7 


2 40 


9 20 


0.9 


1.7 


2.6 


3.5 


4.3 


0.1 


0.2 


0.3 


0.3 


0.4 


0.5 


0.6 


0.7 


O.S 


2 50 


9 10 


0.9 


1.8 


2.7 


3.6 


4.5 


0.1 


0.2 


0.3 


0.4 


0.5 


0.5 


0.6 


0.7 


0.8 


3 


9 


0.9 


1.9 


2.8 


3.8 


4.7 


0.1 


0.2 


0.3 


0.4 


0.5 


0.6 


0.7 


0.7 


0.8 


3 10 


8 50 


1.0 


1.9 


2.9 


3.9 


4.9 


0.1 


0.2 


0.3 


0.4 


0.5 


0.6 


0.7 


0.8 


0.9 


3 20 


8 40 


1.0 2.0 


3.0 


4.0 


5.0 


0.1 


0.2 


0.3 


0.4 


0.5 


0.6 


0.7 


0.8 


0.9 


3 30 


8 30 


1.0 2.1 


3.1 


4.1 


5.2 


0.1 


0.2 


0.3 


0.4 


0.5 


0.6 


0.7 


0.8 


0.9 


3 40 


8 20 


1.1 2.1 


3.2 


4.2 


5.3 


0.1 


0.2 


0.3 


0.4 


0.5 


0.6 


0.7 


0.8 


1.0 


3 50 


8 10 


1.1 


2.2 


3.3 


4.3 


5.4 


0.1 


0.2 


0.3 


0.4 


0.5 


0.7 


0.8 


0.9 


1.0 


4 


8 


1.1 


2.2 


3.3 


4.4 


5.6 


0.1 


0.2 


0.3 


0.4 


0.6 


0.7 


0.8 


0.9 


1.0 


4 10 


7 50 


1.1 


2.3 


3.4 


4.5 


5.7 


0.1 


0.2 


0.3 


0.5 


0.6 


0.7 


0.8 


0.9 


1.0 


4 20 


7 40 


1.2 


2.3 


3.5 


4.6 


5.8 


0.1 


0.2 


0.3 


0.5 


0.6 


0.7 


0.8 


0.9 


1.0 


4 30 


7 30 


1.2 


2.3 


3.5 


4.7 


5.9 


0.1 


0.2 


0.4 


0.5 


0.6 


0.7 


0.8 


0.9 


1.1 


4 40 


7 20 


1.2 


2.4 


3.6 


4.8 


5.9 


0.1 


0.2 


0.4 


0.5 


0.6 


0.7 


0.8 


1.0 


1.1 


4 50 


7 10 


1.2 


2.4 


3.6 


4.8 


6.0 


0.1 


0.2 


0.4 


0.5 


0.6 


0.7 


0.8 


1.0 


1.1 


5 


7 


1.2 


2.4 


3.6 


4.9 


6.1 


0.1 


0.2 


0.4 


0.5 


0.6 


0.Y 


0.9 


1.0 


1.1 


5 10 


6 50 


1.2 


2.5 


3.7 


4.9 


6.1 


0.1 


0.2 


0.4 


0.5 


0.6 


0.7 


0.9 


1.0 


1.1 


5 20 


6 40 


1.2 


2.5 


3.7 


4.9 


6.1 


0.1 


0.2 


0.4 


0.5 


0.6 


0.7 


0.9 


1.0 


1.1 


5 30 


6 30 


1.2 


2.5 


3.7 


5.0 


6.2 


0.1 


0.2 


0.4 


0.5 


0.6 


0.7 


0.9 


1.0 


1.1 


5 40 


6 20 


1.2 


2.5 


3.7 


5.0 


6.2 


0.1 


0.2 


0.4 


0.5 


0.6 


0.7 


0.9 


1.0 


1.1 


5 50 


6 10 


1.2 


2.5 


3.7 


5.0 


6.2 


0.1 


0.2 


0.4 


0.5 


0.6 


0.7 


0.9 


1.0 


1.1 


6 


6 


1.3 


2.6 


3.8 


5.0 


6.3 


!o.i 


0.2 


0.4 


0.5 


0.6 


0.7 


0.9 


1.0 


1.1 



TABLE XCIV. 

Third Differences. 



107 



Time after 






















Time after 


noon or 


10" 


20" 


30" 


40" 


50" 


1' 


2' 


3' 


4' 


5' 


noon or 


midnight. 






















midnight. 


+ 


» 


,. 






» 


-/ 


» 


" 


- 


" 





Oh. 0m. 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 


12h. 0m. 


30 


0.0 


0.1 


0.1 


0.1 


0.2 


0.2 


0.4 


0.5 


0.7 


0.9 


11 30 


1 


0.1 


0.1 


0.2 


0.2 


0.3 


0.3 


0.6 


1.0 


1.3 


1.5 


11 


1 30 


0.1 


0.1 


0.2 


0.3 


0.3 


0.4 


0.8 


1.2 


1.6 


2.1 


10 30 


2 


0.1 


0.2 


0.2 


0.3 


0.4 


0.5 


0.9 


1.4 


1.9 


2.3 


10 


2 30 


0.1 


0.2 


0.2 


0.3 


0.4 


0.5 


1.0 


1.4 


1.9 


2.4 


9 30 


3 


0.1 


0.2 


0.2 


0.3 


0.4 


0.5 


0.9 


1.4 


1.9 


2.3 


9 


3 30 


0.1 


0.1 


0.2 


0.3 


0.4 


0.4 


0.9 


1.3 


1.7 


2.2 


8 30 


4 


0.1 


0.1 


0.2 


0.2 


0.3 


0.4 


0.7 


1.1 


1.5 


1.9 


8 


4 30 


0.0 


0.1 


0.1 


0.2 


0.2 


0.3 


0.6 


0.9 


1.2 


1.5 


7 30 


5 


0.0 


0.1 


0.1 


0.1 


0.2 


0.2 


0.4 


0.6 


08 


1.0 


7 


5 30 


0.0 


0.0 


0.1 


0.1 


0.1 


0.1 


0.2 


0.3 


0.4 


0.5 


6 30 


6 

+ 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 


6 



TABLE XCV. 

Fourth Differences. 



Time after 










ii 






Time after 


noon or 


10" 


20" 


30" 


40" 


50" 


1' 


2' 


3' 


noon or 


midnight. 


















midnight. 


h. m. 


•> 


" 


" 


// 


>, 




« 


» 


h. m. 





0.0 


0.0 


0.0 


00 


0.0 


0.0 


0.0 


0.0 


12 


30 


0.0 


0.1 


0.1 


0.1 


0.2 


0.2 


0.4 


0.6 


11 30 


1 


0.1 


0.1 


0.2 


0.3 


0.3 


0.4 


0.8 


1.2 


11 


1 30 


0.1 


0.2 


0.3 


0.4 


0.5 


| 0.6 


1.2 


1.7 


10 30 


2 


0.1 


0.2 


04 


0.5 


0.6 


0.7 


1.5 


2.2 


10 


2 30 


0.1 


0.3 


0.4 


0.6 


0.7 


0.9 


1.8 


2.7 


9 30 


3 


0.2 


0.3 


0.5 


0.7 


0.9 


1.0 


2.1 


3.1 


9 


3 30 


0.2 


0.4 


0.6 


0.8 


0.9 


1.1 


2.3 


3.4 


8 30 


4 


0.2 


0.4 


0.6 


0.8 


1.0 


1.2 


2.5 


3.7 


8 


4 30 


0.2 


0.4 


0.7 


0.9 


1.1 


1.3 


2.6 


3.9 


7 30 


5 


0.2 


0.5 


0.7 


0.9 


1.1 


1.4 


2.7 


4.1 


7 


5 30 


0.2 


0.5 


! 0.7 


0.9 


1.2 


1.4 


2.8 


4.2 


6 30 





0.2 


0.5 


i 0.7 


0.9 


1.2 


II 1.4 


2.8 


4.2 


6 



103 



TABLE XCVI. Logistical Logarithms. 



/ 

~0 





1 


2 
120 


3 


4 
240 


5 


6 


7 


8 


9 ! 





60 

I 


180 


300 


360 


420 


480 


540 




1.7782 


1.4771 


1.3010 


1.1761 


1.0792 


1.0000 


9331 


8751 


8239 


1 


3.5563 


1.7710 


1.4735 


1.2986 


1.1743 


1.0777 


9988 


9320 


8742 


8231 


2 


3.2553 


1.7639 


1.4699 


1.2962 


1.1725 


1.0763 


9976 


9310 


8733 


8223 


3 


3.0792 


1.7570 


1.4664 


1.2939 


1.1707 


1.0749 


9964 


9300 


8724 


8215 


4 


2.9542 


1.7501 


1.4629 


1.2915 


1.1689 


1.0734 


9952 


9289 


8715 


8207 


5 


2.8573 


1.7434 


1.4594 


1.2891 


1.1671 


1.0720 


9940 


9279 


8706 


8199 


6 


2.7782 


1.7368 


1.4559 


1.2868 


1.1654 


1.0706 


9928 


9269 


8697 


8191 


7 


2.7112 


1.7302 


1.4525 


1.2845 


1.1636 


1.0692 


9916 


9259 


8688 


8183 


8 


2.6532 


1.7238 


1.4491 


1.2821 


1.1619 


1.0678 


9905 


9249 


8679 


8175 


9 


2.6021 


1.7175 


1.4457 


1.2798 


1.1601 


1.0663 


9893 


9238 


8670 


8167 


10 


2.5563 


1.7112 


1.4424 


12775 


1.1584 


1.0649 


9881 


9228 


8661 


8159 


11 


2.5149 


1.7050 


1.4390 


1.2753 


1.1566 


1.0635 


9869 


9218 


8652 


8152 


12 


2.4771 


1.6990 


1.4357 


1 2730 


1.1549 


1.0621 


9858 


9208 


8643 


8144 


13 


2.4424 


1.6930 


1.4325 


1.2707 


1.1532 


1.0608 


9846 


9198 


8635 


8136 


14 


2.4102 


1.6871 


1.4292 


1.2685 


1.1515 


1.0594 


9834 


9188 


8626 


8128 


15 


2.3802 


1.6812 


1.4260 


1.2663 


1.1498 


1.0580 


9823 


9178 


8617 


8120 


16 


2.3522 


1.6755 


1.4228 


1.2640 


1.1481 


1.0566 


9811 


9168 


8608 


8112 


17 


2.3259 


1.6698 


1.4196 


1.2618 


1.1464 


1.0552 


9800 


9158 


8599 


8104 


18 


2.3010 


1.6642 


1.4165 


1.2596 


1.1447 


1.0539 


9788 


9148 


8591 


8097 


19 


2.2775 


1.6587 


1.4133 


1.2574 


1.1430 


1.0525 


9777 


9138 


8582 


8089 


20 


2.2553 


1.6532 


1.4102 


1.2553 


1.1413 


1.0512 


9765 


9128 


8573 


8081 


21 


2. 2341 


1.6478 


1.4071 


1.2531 


1.1397 


1.0498 


9754 


9119 


8565 


8073 


22 


2.2139 


1.6425 


1.4040 


1.2510 


1.1380 


1.0484 


9742 


9109 


8556 


8066 


23 


2.1946 


1.6372 


1.4010 


1.2488 


1.1363 


1.0471 


9731 


9099 


8547 


8058 


24 


2.1761 


1.6320 


1.3979 


1.2467 


1.1347 


1.0458 


9720 


9089 


8539 


8050 


25 


2.15S4 


1.6269 


1.3949 


1.2445 


1.1331 


1.0444 


9708 


9079 


8530 


8043 


26 


2.1413 


1.6218 


1.3919 


1.2424 


1.1314 


1.0431 


9697 


9070 


8522 


8035 


27 


2.1249 


1.6168 


1.3890 


1.2403 


1.1298 


1.0418 


9686 


9060 


8513 


8027 


28 


2.1091 


1.6118 


1.3860 


1.2382 


1.1282 


1.0404 


9675 


9050 


8504 


8020 


29 


2.0939 


1.6069 


1.3831 


1.2362 


1.1266 


1.0391 


9664 


9041 


8496 


8012 


30 


2.0792 


1.6021 


1.3802 


1.2341 


1.1249 


1.0378 


9652 


9031 


8487 


8004 


31 


2.0649 


1.5973 


1.3773 


3.2320 


1.1233 


1.0365 


9641 


9021 


8479 


7997 


32 


2.0512 


1.5925 


1.3745 


1.2300 


1.1217 


1.0352 


9630 


9012 


8470 


7989 


33 


2.0378 


1.5878 


1.3716 


1.2279 


1.1201 


1.0339 


9619 


9002 


8462 


7981 


34 


2.0248 


1.5832 


1.3688 


1.2259 


1.1186 


1.0328 


9608 


8992 


8453 


7974 


35 


2.0122 


1.5786 


1.3660 


1.2239 


1.1170 


1.0313 


9597 


8983 


8445 


7966 


36 


2.0000 


1.5740 


1.3632 


1.2218 


1.1154 


1.0300 


9586 


8973 


8437 


7959 


37 


1.9S81 


1.5695 


1.3604 


1.2198 


1.1138 


1.0287 


9575 


8964 


8428 


7951 


38 


1.9765 


1.5651 


1.3576 


1.2178 


1.1123 


1.0274 


9564 


8954 


8420 


7944 


39 


1.9652 


1.5607 


1.3549 


1.2159 


1.1107 


1.0261 


9553 


8945 


8411 


7936 


40 


1.9542 


1.5563 


1.3522 


1.2139 


1.1091 


1.0248 


9542 


8935 


8403 


7929 


41 


1.9435 


1.5520 


1.3495 


1.2119 


1.1076 


1.0235 


9532 


8926 


8395 


7521 


42 


1.9331 


1.5477 


1.3468 


1.2099 


1.1061 


1.0223 


9521 


8917 


8386 


7914 


43 


1.9228 


1.5435 


1.3441 


1.2080 


1.1045 


1.0210 


9510 


8907 


8378 


7906 


44 


1.9128 


1.5393 


1.3415 


1.2061 


1.1030 


1.0197 


9499 


8898 


8370 


7899 


45 


1.9031 


1.5351 


1.3388 


1.2041 


1.1015 


1.0185 


9488 


8888 


8361 


7891 


46 


1.8935 


1.5310 


1.3362 


1.2022 


1.0999 


1.0172 


9478 


8879 


8353 


7884 


47 


1.8842 


1.5269 


1.3336 


1.2003 


1.0984 


1.0160 


9467 


8870 


8345 


7877 


48 


1.8751 


1.5229 


1.3310 


1.1984 


1.0969 


1.0147 


9456 


8861 


8337 


7869 


49 


1.8661 


1.5189 


1.3284 


1.1965 


1.0954 


1.0135 


9446 


8851 


8328 


7862 


50 


1.8573 


1.5149 


1.3259 


1.1946 


1.0939 


1.0122 


9435 


8842 


8320 


7855 


51 


1.8487 


1.5110 


1.3233 


1.1927 


1.0924 


1.0110 


9425 


8833 


8312 


7847 


52 


1.8403 


1.5071' 


1.3208 


1.1908 


1.0909 


1.0098 


9414 


8824 


8304 


7840 


53 


1.8320 


1.5032 


1.3183 


1.1889 


1.0894 


1.0085 


9404 


8814 


8296 


7832 


54 


1.8239 


1.4994 


1.3158 


1.1871 


1.0880 


1.0073 


9393 


8805 


8288 


7825 


55 


1.8159 


1.4956 


1.3133 


1.1852 


1.0865 


1.0061 


9383 


8796 


8279 


7818 


56 


1.8081 


1.4918 


1.3108 


1.1834 


1.0850 


1.0049 


9372 


8787 


8271 


7811 


57 


1.8004 


1.4881 


1.3083 


1.1816 


1.0835 


1.0036 


9362 


8778 


8263 


7803 


58 


1.7929 


1.4844 


1.3059 


1.1797 


1.0821 


1.0024 


9351 


8769 


8255 


7796 


59 


1.7855 


1.4808 


1.3034 


1.1779 


1.0806 


1.0012 


9341 


8760 


8247 


7789 


60 1.7782 


1 1.4771 


1.3010 


1.1761 1.0792 


1.0000 


9331 


8751 


8239 7782 



TABLE XCVI. Logistical Logarithms. 



109 






10 
600 


Jl 


12 


13 


14 


15 


16 


17 


18 


19 


'20 


21 


660 

7368 


720 
6990 


7S0 


840 
6320 


900 
6021 


960 


1020 

5477 


1080 


1140 


1200 
4771 


1260 
4559 


7782 


6642 


5740 


5229 


4994 


1 


7774 


7361 


6984 


6637 


6315 


6016 


5736 


5473 


5225 


4990 


4768 


4556 


2 


7767 


7354 


6978 


6631 


6310 


6011 


5731 


5469 


5221 


4986 


4764 


4552 


3 


7760 


7348 


6972 


6625 


6305 


6006 


5727 


5464 


5217 


4983 


4760 


4549 


4 


7753 


7341 


6966 


6620 


6300 


6001 


5722 


5460 


5213 


4979 


4757 


4546 


5 


7745 


7335 


6960 


6614 


6294 


5997 


5718 


5456 


5209 


4975 


4753 


4542 


6 


7738 


7328 


6954 


6609 


6289 


5992 


5713 


5452 


5205 


4971 


4750 


4539 


7 


7731 


7322 


6948 


6603 


6284 


5987 


5709 


5447 


5201 


4967 


4746 


4535 


8 


7724 


7315 i 


6942 


6598 


6279 


5982 


5704 


5443 


5197 


4964 


4742 


4532 


9 


7717 


7309 


6936 


6592 


6274 


5977 


5700 


5439 


5193 


4960 


4739 


4528 


10 


7710 


7002 


6930 


6587 


6269 


5973 


5695 


5435 


5189 


4956 


4735 


4525 


11 


7703 


/296 


6924 


6581 


6264 


5968 


5691 


5430 


5185 


4952 


4732 


4522 


12 


7696 . 


7289 


6918 


6576 


6259 


5963 


5686 


5426 


5181 


4949 


4728 


4518 


13 


7688 


7283 


6912 


6570 


6254 


5958 


5682 


5422 


5177 


4945 


4724 


4515 


14 


7681 


7276 


6906 


6565 


6248 


5954 


5677 


5418 


5173 


4941 


4721 


4511 


15 


7674 


7270 


6900 


6559 


6243 


5949 


5673 


5414 


5169 


4937 


4717 


4508 


16 


7667 


'.'264 


6894 


6554 


6238 


5944 


5669 


5409 


5165 


4933 


4714 


4505 


17 


7660 


7257 


6888 


6548 


6233 


5939 


5664 


5405 


5161 


4930 


4710 


4501 


18 


7653 


7251 


6882 


6543 


6228 


5935 


5660 


5401 


5157 


4926 


4707 


4498 


19 


7646 


7244 


6877 


6538 


6223 


5930 


5655 


5397 


5153 


4922 


4703 


4494 


20 


7639 


7238 


6871 


6532 


6218 


5925 


5651 


5393 


5149 


4918 


4699 


4491 


21 


7632 


7232 


6865 


6527 


6213 


5920 


5646 


5389 


5145 


4915 


4696 


4488 


22 


7625 


7225 


6859 


6521 


6208 


5916 


5642 


5384 


5141 


4911 


4692 


4484 


23 


7618 


7219 


6853 


6516 


6203 


5911 


5637 


5380 


5137 


4907 


4689 


4481 


24 


7611 


7212 


6847 


6510 


6198 


5906 


5633 


5376 


5133 


4903 


4685 


4477 


25 


7604 


7206 


6841 


6505 


6193 


5902 


5629 


5372 


5129 


4900 


4682 


4474 


26 


7597 


7200 


6836 


6500 


6188 


5897 


5624 


5368 


5125 


4896 


4678 


4471 


27 


7590 


7193 


6830 


6494 


6183 


5892 


5620 


5364 


5122 


4892 


4675 


4467 


28 


7583 


7187 


6824 


6489 


6178 


5888 


5615 


5359 


5118 


4889 


4671 


4464 


29 


7577 


7181 


6818 


6484 


6173 


5883 


5611 


5355 


5114 


4885 


466 8 


4460 


30 


7570 


7175 


6812 


6478 


6168 


5878 


5607 


5351 


5110 


4881 


4664 


4457 


31 


7563 


7168 


6S07 


6473 


6163 


5874 


5602 


5347 


5106 


4877 


4660 


4454 


32 


7556 


7162 


6801 


6467 


6158 


5869 


5598 


5343 


5102 


4874 


4657 


4450 


33 


7549 


7156 


6795 


6462 


6153 


5864 


5594 


5339 


5098 


4870 


4653 


4447 


34 


7542 


7149 


6789 


6457 


6148 


5860 


5589 


5335 


5094 


4866 


4650 


4444 


35 


7535 


7143 


6784 


6451 


6143 


5855 


5585 


5331 


5090 


4863 


4646 


4440 


36 


7528 


7137 


6778 


6446 


6138 


5850 


5580 


5326 


5086 


4859 


4643 


4437 


37 


7522 


7131 


6772 


6441 


6133 


5846 


5576 


5322 


5082 


4855 


4639 


4434 


38 


7515 


7124 


6766 


6435 


6128 


5841 


5572 


5318 


5079 


4852 


4636 


4430 


39 


7508 


7118 


6761 


6430 


6123 


5836 


5567 


5314 


5075 


:4S48 


4632 


4427 


40 


7501 


7112 


6755 


6425 


6118 


5832 


5563 


5310 


5071 


4844 


4629 


4424 


41 


7494 


7106 


6749 


6420 


6113 


5827 


5559 


5306 


5067 


4841 


4625 


4420 


42 


[7488 


7100 


6743 


6414 


6108 


5823 


5554 


5302 


5063 


4837 


4622 


4417 


43 


'7481 


7093 


6738 


6409 


6103 


5818 


5550 


5298 


5059 


4833 


4618 


4414 


44 


'7474 


7087 


6732 


6404 


6099 


5813 


5546 


5294 


5055 


4830 


4615 


4410 


45 


7467 


7081 


6726 


6398 


6094 


5809 


5541 


5290 


5051 


4826 


4611 


4407 


46 


7461 


7075 


6721 


6393 


60S9 


5804 


5537 


5285 


5048 


4822 


4608 


4404 


47 


7454 


7069 


6715 


6388 


6084 


5800 


5533 


5281 


5044 


4819 


14604 


4400 


48 


7447 


7063 


6709 


6383 


6079 


5795 


5528 


5277 


5040 


4815 


4601 


4397 


49 


:7441 


7057 


6704 


6377 


6074 


5790 


5524 


5273 


5036 


4811 


4597 


4394 


50 1 7434 


7050 


669S 


6372 


6069 


5786 


5520 


5269 


5032 


4808 


4594 


4390 


51 1 7427 


7044 


6692 


6367 


6064 


5781 


5516 


5265 


5028 


4804 


4590 


4387 


52 • 7421 


7038 


6687 


6362 


6059 


5777 


5511 


5261 15025 


'4800 


4587 


4384 


53 


7414 


7032 


6681 


6357 


; 6055 


5772 


5507 


15257 5021 


4797 


4584 


4380 


54 


7407 


7026 


6676 


6351 


16050 


5768 


j 5503 


5253 5017 


4793 


4580 


4377 


55 


7401 


7020 


6670 


6346 


6045 


5763 


5498 


5249 


5013 


4789 


'4577 


4374 


56 


7394 


7014 


6664 


6341 


6040 


5758 


5494 


5245 


5009 


4786 | 4573 


4370 


57 


7387 


7008 


6659 


6336 


6035 


5754 


5490 


5241 


5005 


4782 


4570 


4367 


58 


7381 


7002 


6653 


6331 


6030 


J5749 


5486 


5237 


5002 


4778 


4566 


4364 


59 


7374 


6996 


6648 


6325 


6025 


1 5745 


5481 


5233 


j 4998 


4775 


4563 


4361 


60 


7368 


6990 


6642 


6320 


6021 


15740 


5477 


5229 


1 4994 


4771 


4559 


4357 



no 



TABLE XCVI. 



Logistical Logarithms. 






22 i 


23 J 


24 


iii 

1500 

3302| 


26 
7560 
3632 


27 | 28 


29 


30 
1800 


JL\ 


32 


33 


1320 


1380 1440 


1620 


1680 


1740 


I860 
2868 | 


1920 


IdSO 


4357 ; 4164 


3979 


3468 


3310 


3158 


3010 


2730 


2596 


1 


4354 4161 


3976 


3799 ! 


3829 


3465 


3307 


3155 


300S 


2866 


2728 


2594 


2 


4351 j 4158 


3973 


3796 i 


3626 


3463 


3305 


3153 


3005 


2863! 


2725 


2592 


3 


4347 1 4155 


3970 


3793 


3623 


3460 


3302 


3150 


3003 


2861 1 


2723 


2590 


4 


4344 I 4152 


3967 


3791: 


3621 


3457 


3300 


3148 


3001 


2859 j 


2721 


2588 


5 


4341 : 4149 


3964 


37S3 


3618 


3454 


3297 


3145 


2998 


2856| 


2719 


2585 


6 


4338 ! 4145 


3961 


3785 : 


3615 


3452 


3294 


3143 


2996 


2854 


2716 


2583 


7 


4334 i 4142 


3958 


3782 


3612 


3449 


3292 


3140 


2993 


2352 


2714 


2581 


8 


4331 4139 


3955 


3779 


3610 


3446 


3289 


3138 


2991 


28-! 9 


2712 


2579 


9 


4328 4136 


3952 


3776 


3607 


3444 


3287 


3135 


2989 


284/ ' 


2710 


2577 


10 


4325 \ 4133 


3949 


3773 


3604 


3441 


3284 


3133 


2986 


2845 2707 


2574 


11 


4321 i 4130 


3946 


3770' 


3601 


3438 


3282 


3130 


2984 


2842 


2705 


2572 


12 


4318 ; 4127 


3943 


3768 


3598 


3436 


3279 


3128 


2981 


2840 


2703 


2570 


13 


4315 i 4124 


3940 


3765 


3596 


3433 


3276 


3125 


2979 


2838 


2701 


2568 


14 


4311 i 4120 


3937 


3762 


3593 


3431 


3274 


3123 


2977 


2835 


2698 


2566 


15 


4308 j 4117 


3934 


3759 


3590 


3428 


3271 


3120 


2974 


2833 


2696 


2564 


16 


4305 | 4114 


3931 


3756 


3587 


3425 


3269 


3118 


2972 


2831 


2694 


2561 


17 


4302 14111 


392S 


3753 


3585 


342^3 


3266 


3115 


2969 


2828 


2692 


2559 


18 


4298 ! 4108 


3925 


3750 


3532 


3420 


3264 


3113 


2967 


2826 


2689 


2557 


19 


4295 1 4105 


3922 


3747 


3579 


3417 


3261 


3110 


2965 


2824 


2687 


2555 


20 


4292 4102 


3919 


3745 


3576 


341 £ 


3259 


3108 


2962 


2821 


2635 


2553 


21 


4289 


4099 


3917 


3742 


3574 


3412 


3256 


3105 


2960 


2819 


2683 


2551 


22 


4285 


4096 


3914 


3739 


3571 


3409 


3253 


3103 


2958 


2817 


2681 


2548 


23 


4282 


4092 


3911 


3736 


3568 


3407 


3251 


3101 


2955 


2815 


2678 


2546 


24 


4279 


4089 


3908 


3733 


3565 


3404 


3243 


3098 


2953 


2812 


2676 


2544 


25 


4276 


4086 


3905 


3730 


3563 


3401 


3246 


3096 


2950 


2810 


2674 


2542 


26 


4273 


4083 


3902 


3727 


3560 


3399 


3243 


3093 


2948 


2808 


2672 


2540 


27 


4269 


4080 


3399 


3725 


3557 


3396 


3241 


3091 


2946 


2805 


2669 


2538 


2S 


4266 


4077 


3896 


3722 


3555 


3393 


3238 


3088 


2943 


2803 


2667 


2535 


29 


4263 


4074 


3893 


3719 


3552 


3391 


3236 


3036 


2941 


2801 


2665 


2533 


30 


4260 


4071 


3890 


3716 


3549 


3388 


3233 


30S3 


2939 


2798 


2663 


2531 


31 


4256 


4068 


3SS7 


3713 


3546 


33S6 


3231 


3081 


2936 


2796 


2660 


2529 


32 


4253 


4065 


3884 


3710 


3544 


3383 


3228 


3078 


2934 


2794 


2658 


2527 


33 


4250 


40 52 


38S1 


3708 


3541 


3330 


3225 


3076 


2931 


2792 


2656 


2525 


34 


4247 


4059 


3878 


3705 


3533 


3373 


3223 


3073 


2929 


2789 


2654 


2522 


35 


4244 


4055 


3875 


3702 


3535 


3375 


3220 


3071 


2927 


2787 


2652 


2520 


36 


4240 


4052 


3872 


3699 


3533 


3372 


3218 


3069 


2924 


2785 


2649 


2518 


37 


4237 


4049 


3869 


3696 


3530 


3370 


3215 


3066 


2922 


2782 


2647 


2516 


3S 


4234 


4046 


3866 


3693 


3527 


3367 


3213 


3064 


2920 


2780 


2645 


2514 


39 


4231 


4043 


3S63 


3691 


3525 


3365 


3210 


3061 


2917 


2778 


2643 


2512 


40 


4228 


4040 


3860 


36S8 


3522 


3362 


320S 


3059 


2915 


2775 


2640 


2510 


41 


4224 


4037 


3857 


3685 


3519 


3359 


3205 


3056 


2912 


2773 


2638 


2507 


42 


4221 


4034 


3855 


3682 


3516 


3357 


3203 


3054 


2910 


2771 


2636 


2505 


43 


42 IS 


4031 


3852 


3679 


3514 


3354 


3200 


3052 


2903 


2769 


2634 


2503 


44 


4215 


402S 


3849 


13677 


3511 


3351 


3198 


3049 


2905 


2766 


2632 


2501 


45 


4212 


4025 


3846 


3674 


3508 


3349 


3195 


13047 


2903 


2764 


2629 


2499 


46 


4209 


4022 


3843 


3671 


3506 


3346 


3193 


3044 


2901 


2762 


2627 


2497 


47 


4205 


4019 


3S40 


3663 


3503 


3344 


3190 


3042 


2898 


2760 


2625 


2494 


48 


4202 


4016 


3837 


3665 


3500 


3341 


3183 


3039 


2896 


2757 


2623 


2492 


49 


4199 


4013 


3834 


3663 


3497 


3338 


3185 


3037 


2894 


2755 


2621 


2490 


50 


4196 


4010 


3831 


3660 


3495 


3336 


3183 


3034 


2891 


2753 


2618 


2488 


51 


4193 


4007 


3328 


3657 


3492 


3333 


3180 


3032 


2889 


2750 


2616 


2486 


52 


41S9 


4004 


3825 


j 3654 


3489 


3331 


3178 


3030 


2S87 


2748 


2614 


2484 


53 


41S6 


4001 


3822 


3651 


! 3487 


3328 


3175 


3027 


2884 


2746 


2612 


2482 


54 


4183 


399S 3820 3649 


34S4 


3325 


3173 


3025 


2882 


2744 


2610 


2480 


55 


4180 


3995 3S17 


3646 


13481 


3323 


3170 


13022 


2880 


2741 


2607 


2477 


56 


4177 


3991 


3814 


3643 


3479 


3320 


3168 


1 3020 


2S77 


2739 


2605 


2475 


57 


4174 


3988 


3311 


3640 


13476 


3318 


3165 


3018 


2875 


2737 ! 2603 


2473 


58 


4171 


3985 


3893 


3637 


' 3473 


3315 


3163 


3015 


,2873 


2735 2601 


2471 


59 


4167 


3982 


3S05 


3635 


3471 


3313 


3160 


3013 


J2S70 


2732 J 2599 


2469 


60 
I 


4164 


3979 


3802 


3632 346S 


3310 


3158 


3010 


12868 


2730 1 2596 


2467 



TABLE XCVJ. Logistical Logarithms. 



Ill 





34 
2040 


35 
a 100 


36 


37 


38 j 


39 


40 | 
2400 
1761 


41 
2460 


42 
"2520 


-ill 


44 


45 ! 


2160 


2220 
2099 


2280 2340 


2580j 


2640 


2700 


2467 


2311 


22 IS 


1984 


1871 


1654 


1549 


1447 


1347 


"1249 


1 


2465 


2339 


2216! 


209S 


19S2 


1S69 


1759 


1652 


1547 


1445 


1345 


1248 


•J 


2462 


2337 


2214 


2096 


19S0 


1867 


1757 


1650 


1546 


1443 


1344 


1246 


i 3 


2460 


2335 


2212 


2094 


1978 


1865 


1755 


1648 


1544 


1442 


1342 


1245 


i 4 


2458 


2333 


2210 


2092 


1976 


1863 


1754 


1647 


1542 


1440 


1340 


1243 


1 5 


2456 


2331 


2208 


2090 


1974 


1862 


17&2 


1645 


1540 


1438 


1339 


1241 


6 


2454 


2328 


2206 


2088 


1972 


1860 


1750 


1643 


1539 


1437 


1337 


1240 


7 


2452 


2326 


2204 


20S6 


1970 


1858 


1748 


1641 


1537 


1435 


1335 


1238 


8 


2450 


2324 


2202 


20S4 


196S 


1856 


1746 


1640 


1535 


1433 


1334 


1237 


9 


2448 


2322 


2200 


2082 


1967 


1854 


1745 


1638 


1534 


1432 


1332 


1235 


10 


2445 


2320 


2198 


2080 


1965 


1S52 


1743 


1636 


1532 


1430 


1331 


1233 


11 


2443 


2318 


2196 


2078 


1963 


1S50 


1741 


1634 


1530 


1428 


1329 


1232 


12 


2441 


2316 


2194 


2076 


1961 


1849 


1739 


1633 


1528 


1427 


1327 


1230 


13 


2439 


2314 


2192 


2074 


1959 


1847 


1737 


1631 


1527 


1425 


1326 


1229 


14 


343V 


2312 


2190 


2072 


1957 


1845 


1736 


1629 


1525 


1423 


1324 


1227 


15 


2435 


2310 


2188 


2070 


1955 


1S43 


1734 


1627 


1523 


1422 


1322 


1225 


16 


2433 


2308 


21S6 


2068 


1953 


1841 


1732 


1626 


1522 


1420 


1321 


1224 


17 


2431 


2306 


21S4 


2066 


1951 


1839 


1730 


1624 


1520 


1418 


1319 


1222 


18 


2429 


2304 


2182 


2064 


1950 


1838 


1728 


1622 


1518 


1417 


1317 


1221 


19 


2426 


2302 


2180 


2062 


1948 


1836 


1727 


1620 


1516 


1415 


1316 


1219 


20 


2424 


2300 


2178 


2061 


1946 


1834 


1725 


1619 


1515 


1413 


1314 


1217 


21 


2422 


2298 


2176 


2059 


1944 


1832 


1723 


1617 


1513 


1412 


1313 


1216 


22 


2420 


2296 


2174 


2057 


1942 


1830 


1721 


1615 


1511 


1410 


1311 


1214 


23 


241S 


2294 


2172 


2055 


1940 


1828 


1719 


1613 


1510 


1408 


1309 


1213 


24 


2416 


2291 


2170 


2053 


1938 


1S27 


1718 


1612 


1508 


1407 


1308 


1211 


25 


2414 


22S9 


2169 


2051 


1936 


1825 


1716 


1610 


1506 


1405 


1306 


1209 


26 


2412 


2287 


2167 


2049 


1934 


1823 


1714 


1608 


1504 


1403 


1304 


1208 


27 


2410 


2285 


2165 


2047 


1933 


1821 


1712 


1606 


1503 


1402 


1303 


1206 


28 


2408 


2283 


2163 


2045 


1931 


1819 


1711 


1605 


1501 


1400 


1301 


1205 


29 


2405 


2281 


2161 


2043 


1929 


1817 


1709 


1603 


1499 


139S 


1300J1203 


30 


2403 


2279 


2159 


2041 


1927 


1816 


1707 


1601 


1498 


1397 


1298 


1201 


31 


2401 


2277 


2157 


2039 


1925 


1814 


1705 


1599 


1496 


1395 


1296 


1200 


32 


2399 


2275 


2155 


2037 


1923 


1812 


1703 


1598 


1494 


1393 


1295 


1198 


33 


2397 


2273 


2153 


2035 


1921 


1810 


1702 


1596 


1493 


1392 


1293 


1197 


34 


2395 


2271 


2151 


2033 


1919 


1S0S 


1700 


1594 


1491 


1390 


1291 


1195 


35 


2393 


2269 


2149 


2032 


1918 


1806 


1698 


1592 


1489 


13S8 


1290 


1193 


36 


2391 


2267 


2147 


2030 


1916 


1805 


1696 


1591 


1487 


1387 


1288 


1192 


37 


2389 


2265 


2145 


2028 


1914 


1803 


1694 


1589 


I486 


1385 


12S7 


1190 


38 


2387 


2263 


2143 


2026 


1912 


1801 


1693 


1587 


1484 


1383 


12S5 


1189 


39 


23S4 


2261 


2141 


2024 


1910 


1799 


1691 


15S5 


1482 


1382 


12S3I1187 


40 


23S2 


2259 


2139 


2022 


1908 


1797 


1689 


1584 


14S1 


1380 


1282 1186 


41 


23S0 


2257 


2137 


2020 


1906 


1795 


1687 


15S2 


1479 


1378 


1280 


1184 


42 


2378 


2255 


2135 


2018 


1904 


1794 


1686 


1580 


1477 


1377 


1278 


1182 


43 


2376 


2253 


2133 


2016 


1903 


1792 


1684 


1578 


1476 


1375 


1277 


1181 


44 


2374 


2251 


2131 


2014 


1901 


1790 


1682 


1577 


1474 


1373 


1275 


1179 


45 


2372 


2249 


2129 


2012 


1899 


178S 


1680 


1575 


1472 


1372 


1274 


1178 


46 


2370 


2247 


2127 


2010 


1897 


1786 


1678 


1573 


1470 


1370 


1272 


1176 


47 


2368 


2245 


2125 


2009 


1895 


1785 


1677 


1571 


1469 


1363 


1270 


1174 


|48 


2366 


2243 


2123 


2007 


1S93 


1783 


1675 


1570 


1467 


1367 


1269 


1173 


49 


2364 


2241 


2121 


2005 


1891 


1781 


1673 


1568 


1465 


1305 


1267 


1171 


50 


2362 


2239 


2119 


2003 


1889 


1779 


1671 


1566 


1464 


1363 


1266 


1170 


51 


2359 


2237 


2117 


2001 


1888 


1777 


1670 


1565 


1462 


1362 


1264 1168 


52 


2357 


2235 


2115 


1999 


1886 


1775 1668 


1563 


1460 


1360 


1262 1167 


53 


2355 


2233 


2113 


1997 


1884 


1774 1666 


1561 


1459 


1359 


1261 1165 


54 


2353 


2231 


2111 


1995 


1882 


1772 


1664 


1559 


1457 


1357 


1259 ! 1163 


55 


2351 


2229 


2109 


1993 


1S80 


1770 


1663 


1558 


1455 


1355 


1257 1162 


56 


2349 


2227 


2107 


1991 


1878 


1768 


1661 


1556 


1454 


1354 


1256 


1160 


57 


2347 


2225 


2105 


19S9 


1876 


1766 


1659 


1554 


1452 


1352 


1254 


1159 


58 


2345 


2223 


2103 


1987 


1875 


1765 


1657 


1552 


1450 


1350 


1253 


1157 


59 


2343 


2220 


2101 


1986 


1873 


1763 


1655 


1551 


1449 


1349 


1251 


1156 


60 


2341 


2218 


2099 1984 


1871 


1761 


1654 


1549 ' 1447 ! 1347 


1249 


1154 



112 



TABLE XCVI. Logistical Logarithms. 



r~, — 


46 


47 


48 
2880 
0969 


49 


50 


51 


52 


53 
3180 
0539 


54 
3240 
0458 


55 


56 


57 


58 


59 


i " 


2760 


2820 


2940 

0880 


3000 


3060 


3120 
0621 


3300 
0378 


3360 


3420 


3480 


3540 


i~° 


1154 


1061 


0792 


0706 


0300 


0223 


0147 


0073 


i 1 


1152 


1059 


0968 


0878 


0790 


0704 


0620 


0537 


0456 


0377 


0298 


0221 


0146 


0072 


2 


1151 


1057 


0966 


0S77 


0789 


0703 


0619 


0536 


0455 


0375 


0297 


0220 


0145 


0071 


3 


1149 


1056 


0965 


0875 


07S7 


0702 


0617 


0535 


0454 


#374 


0296 


0219 


0143 


0069 


4 


1148 


1054 


0963 


0S74 


0786 


0700 


0616 


0533 


0452 


0373 


0294 


0218 


0142 


0068 


5 


1146 


1053 


0962 


0872 


0785 


0699 


0615 


0532 


0451 


0371 


0293 


0216 


0141 


0067 


i 6 


1145 


1051 


0960 


0871 


0783 


0697 


0613 


0531 


0450 


0370 


0292 


0215 


0140 


0066 


7 


1143 


1050 


0859 


0S69 


0782 


0696 


0612 


0529 


0448 


0369 


0291 


0214 


0139 


0064 


8 


1141 


1048 


0957 


0868 


0780 


0694 


0610 


0528 


0447 


0367 


0289 


0213 


0137 


0063 


! 9 


1140 


1047 


0956 


0866 


0779 


0693 


0609 


0526 


0446 


0366 


0288 


0211 


0136 


0062 


10 


1138 


1045 


0954 


0S65 


0777 


0692 


0608 


0525 


0444 


0365 


0287 


0210 


0135 


0061 


ill 


1137 


1044 


0953 


0863 


0776 


0690 


0606 


0524 


0443 


0363 


0285 


0209 


0134 


0060 


12 


1135 


1042 


0951 


0S62 


0774 


0689 


0605 


0522 


0442 


0362 


0284 


0208 


0132 


0058 


13 


1134 


1041 


0950 


0360 


0773 


0687 


0603 


0521 


0440 


0361 


0283 


0206 


0131 


0057 


14 


1132 


1039 


0948 


0S59 


0772 


0686 


0602 


0520 


0439 


0359 


0282 


0205 


0130 


0056 


15 


1130 


1037 


0947 


0857 


0770 


0685 


0601 


0518 


0438 


0358 


0280 


0204 


0129 


0055 


16 


1129 


1036 


0945 


0856 


0769 


0683 


0599 


0517 


0436 


0357 


0279 


0202 


0127 


0053 


'17 


1127 


1034 


0944 


0855 


0767 


0682 


0593 


0516 


0435 


0356 


0278 


0201 


0126 


0052 


]18 


1126 


1033 


0942 


0853 


0766 


0680 


0596 


0514 


0434 


0354 


0276 


0200 


0125 


0051 


19 


1124 


1031 


0941 


0S52 


0764 


0679 


0595 


0513 


0432 


0353 


0275 


0199 


0124 


0050 


20 


1123 


1030 


0939 


0S50 


0763 


0678 


0594 


0512 


0431 


0352 


0274 


0197 


0122 


0049 


21 


1121 


1028 


093S 


0849 


0762 


0676 


0592 


0510 


0430 


0350 


0273 


0196 


0121 


0047 


22 


1119 


1027 


0936 


0S47 


0760 


0675 


0591 


0509 


0428 


0349 


0271 


0195 


0120 


0046 


23 


1118 


1025 


0935 


0846 


0759 


0673 


0590 


0507 


0427 


0348 


0270 


0194 


0119 


0045 


24 


1116 


1024 


0933 


0844 


0757 


0672 


0588 


0506 


0426 


0346 


0269 


0192 


0117 


0044 


25 


1115 


1022 


0932 


0843 


0756 


0670 


0587 


0505 


0424 


0345 


0267 


0191 


0116 


0042 


26 


1113 


1021 


0930 


0841 


0754 


0669 


05S5 


0503 


0423 


0344 


0266 


0190 


0115 


0041 


27 


1112 


1019 


0929 


0840 


0753 


0668 


0584 


0502 


0422 


0342 


0265 


0189 


0114 


0040 


28 


1110 


1018 


0927 


0838 


0751 


0666 


0583 


0501 


0420 


0341 


0264 


0187 


0112 


0039 


29 


1109 


1016 


092G 


0S37 


0750 


0665 


0581 


0499 


0419 


0340 


0262 


01S6 


0111 


0038 


30 


1107 


1015 


0924 


0835 


0749 


0663 


0580 


0498 


0418 


0339 


0261 


0185 


0110 


0036 


31 


1105 


1013 


0923 


0834 


0747 


0662 


0579 


0497 


0416 


0337 


0260 


0184 


0109 


0035 


32 


1104 


1012 


0921 


0833 


0746 


0661 


0577 


0495 


0415 


0336 


0258 


0182 


0107 


0034 


33 


1102 


1010 


0920 


0S31 


0744 


0659 


0576 


0494 


0414 


0335 


0257 


0181 


0106 


0033 


34 


1101 


1008 


0918 


0330 


0743 


0658 


0574 


0493 


0412 


0333 


0256 


0180 


0105 


0031 


35 


1099 


1007 


0917 


0828 


0741 


0656 


0573 


0491 


0411 


0332 


0255 


0179 


0104 


0030 


36 


1098 


1005 


0915 


0327 


0740 


0655 


0572 


0490 


0410 


0331 


0253 


0177 


0103 


0029 


37 


1096 


1004 


0914 


0825 


0739 


0654 


0570 


0489 


0408 


0329 


0252 


0176 


0101 


002S 


38 


1095 


1002 


0912 


0824 


0737 


0652 


0569 


0487 


0407 


0328 


0251 


0175 


0100 


0027 


39 


1093 


1001 


0911 


0822 


0736 


0651 


0568 


0486 


0406 


0327 


0250 


0174 


0099 


0025 


40 


1091 


0999 


0909 


0321 


0734 


0649 


0566 


0484 


0404 


0326 


024S 


0172 


0098 


0024 


41 


1090 


0998 


0908 


0819 


0733 


0648 


0565 


0483 


0403 


0324 


0247 


0171 


0096 0023 


42 


1088 


0996 


0906 


0818 


0731 


0047 


0563 


0482 


0402 


0323 


0246 


0170 


0095 0022 


43 


1087 


0995 


0905 


0816 


0730 


0645 


0562 


0480 


0400 


0322 


0244 


0169 


00941 0021 


44 


1085 


0993 


0903 


0815 


0729 


0644 


0561 


0479 


0399 


0320 


0243 


0167 


0093 0019 


45 


1084 


0992 


0902 


0814 


0727 


0642 


0559 


0478 


0398 


0319 


0242 


0166 


0091 


0018 


46 


1082 


0990 


0900 


6812 


0726 


0641 


055S 


0476 


0396 


0318 


0241 


0165 


0090 


0017 


47 


1081 


09S9 


0899 


0811 


0724 


0640 , 0557 


0475 


0395 


0316 


0239 


0163 


0089 


0016 


48 


1079 


0987 


0897 


0809 


0723 


0638 0555 


0474 0394 


0315 


0238 


0162 


0088 


0015 


49 


1078 


0986 


0896 


0808 0721 


0637 0554 


0472 i 0392 


0314 


0237 


0161 


0087 


0013 


60 


1076 


0984 


0894 


0S06 | 0720 


0635 j 0552 


0471 


0391 


0313 


0235 


0160 


0085 


0012 


51 


1074 


0983 


0893 


0805 0719 


0634 ' 0551 


0470 


0390 


0311 


0234 


0158 


0084 0011} 


62 


1073 


0981 


0891 


0803 | 0717 


0633 


0550 


0468 


03S8 


0310 


0233 


0157 


0083 0010 | 


S3 


1071 


0980 


0890 


0S02 0716 


0631 


0548 


0467 


0387 


0309 


0232 


0156 


0082| 0003 


54 


1070 


0978 


0SS8 


0801 !0714 


0630 


0547 


0466 


0386 


0307 


0230 


0155 


00S0 0007 J 


55 


1068 J 0977 | 0887 


0799 |0713 10628 


0546 


0464 


0384 0306 


0229 


0153 


0079 


oooe; 


56 


1067 


0975 


0885 


0798 0711 0627 


0544 


0463 


03S3 


0305 


02*28 


0152 


0078 


0005 


?57 


1065 


0974 


0884 


0796 0710 0626 


0543 


0462 


0382 


0304 


0227 


0151 


0077 


0004 


08 


1064 


0972 


0883 


0795 


0709 


0624 


0541 


0460 


0381 


0302 


0225 


0150 


0075 


0002 


59 


1062 


0971 


0881 


0793 


0707 


0623 


0540 


0459 


0379 


0301 


0224 


0148 


0074 


0001 


A0 


1061 


0969 


0880 


0792 


0706 


0621 10539 0458 


0378 


0300 | 0223 


0147 0073 


0000 



